# 概率密度函数

Boxplot and probability density function of a normal distribution N(0, σ2). 正态分布的箱线图和概率密度函数 N(0, σ2)

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.[1] In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there are an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there are an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1.

In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1.

The terms "probability distribution function"[2] and "probability function"[3] have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion.[4] In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.

The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.

"概率分布函数"[2] 和 "概率函数"[3] 两个词有时也被用来表示概率密度函数。然而，这种用法在概率论统计学领域中并不标准。在其他资料中，当概率分布被定义为一般数值集上的函数时，可以使用 "概率分布函数"这个词，或者它指的也可以是累积分布函数，或者它可以是概率质量函数（PMF）而不是密度。而"密度函数"本身也被用于概率质量函数，这导致了进一步的混淆。[4] 不过一般来说，PMF是在离散型随机变量（在可数集上取值的随机变量）的背景下使用的，而PDF是在连续型随机变量的背景下使用的。

## Example 示例

Suppose bacteria of a certain species typically live 4 to 6 hours. The probability that a bacterium lives 模板:Em 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.0000000000... hours. However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02 (i.e., 2%). Then, the probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, and so on.

Suppose bacteria of a certain species typically live 4 to 6 hours. The probability that a bacterium lives 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.0000000000... hours. However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02 (i.e., 2%). Then, the probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, and so on.

In these three examples, the ratio (probability of dying during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour−1). For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and (0.02 probability / 0.01 hours) = 2 hour−1. This quantity 2 hour−1 is called the probability density for dying at around 5 hours. Therefore, the probability that the bacterium dies at 5 hours can be written as (2 hour−1) dt. This is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where dt is the duration of this window. For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour−1)×(1 nanosecond) ≈ 模板:Val (using the unit conversion 模板:Val nanoseconds = 1 hour).

In these three examples, the ratio (probability of dying during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour−1). For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and (0.02 probability / 0.01 hours) = 2 hour−1. This quantity 2 hour−1 is called the probability density for dying at around 5 hours. Therefore, the probability that the bacterium dies at 5 hours can be written as (2 hour−1) dt. This is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where dt is the duration of this window. For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour−1)×(1 nanosecond) ≈ (using the unit conversion nanoseconds = 1 hour).

There is a probability density function f with f(5 hours) = 2 hour−1. The integral of f over any window of time (not only infinitesimal windows but also large windows) is the probability that the bacterium dies in that window.

There is a probability density function f with f(5 hours) = 2 hour−1. The integral of f over any window of time (not only infinitesimal windows but also large windows) is the probability that the bacterium dies in that window.

## Absolutely continuous univariate distributions 绝对连续的单变量分布

A probability density function is most commonly associated with absolutely continuous univariate distributions. A random variable $\displaystyle{ X }$ has density $\displaystyle{ f_X }$, where $\displaystyle{ f_X }$ is a non-negative Lebesgue-integrable function, if:

A probability density function is most commonly associated with absolutely continuous univariate distributions. A random variable $\displaystyle{ X }$ has density $\displaystyle{ f_X }$, where $\displaystyle{ f_X }$ is a non-negative Lebesgue-integrable function, if:

$\displaystyle{ \Pr [a \le X \le b] = \int_a^b f_X(x) \, dx . }$

Hence, if $\displaystyle{ F_X }$ is the cumulative distribution function of $\displaystyle{ X }$, then:

Hence, if $\displaystyle{ F_X }$ is the cumulative distribution function of $\displaystyle{ X }$, then:

$\displaystyle{ F_X(x) = \int_{-\infty}^x f_X(u) \, du , }$

and (if $\displaystyle{ f_X }$ is continuous at $\displaystyle{ x }$)

and (if $\displaystyle{ f_X }$ is continuous at $\displaystyle{ x }$)

$\displaystyle{ f_X(x) = \frac{d}{dx} F_X(x) . }$

Intuitively, one can think of $\displaystyle{ f_X(x) \, dx }$ as being the probability of $\displaystyle{ X }$ falling within the infinitesimal interval $\displaystyle{ [x,x+dx] }$.

Intuitively, one can think of $\displaystyle{ f_X(x) \, dx }$ as being the probability of $\displaystyle{ X }$ falling within the infinitesimal interval $\displaystyle{ [x,x+dx] }$.

## Formal definition 正式定义

(This definition may be extended to any probability distribution using the measure-theoretic definition of probability.)

(This definition may be extended to any probability distribution using the measure-theoretic definition of probability.)

(这个定义可以借助概率的测度理论定义扩展到任何概率分布。）

A random variable $\displaystyle{ X }$ with values in a measurable space $\displaystyle{ (\mathcal{X}, \mathcal{A}) }$ (usually $\displaystyle{ \mathbb{R}^n }$ with the Borel sets as measurable subsets) has as probability distribution the measure XP on $\displaystyle{ (\mathcal{X}, \mathcal{A}) }$: the density of $\displaystyle{ X }$ with respect to a reference measure $\displaystyle{ \mu }$ on $\displaystyle{ (\mathcal{X}, \mathcal{A}) }$ is the Radon–Nikodym derivative:

A random variable $\displaystyle{ X }$ with values in a measurable space $\displaystyle{ (\mathcal{X}, \mathcal{A}) }$ (usually $\displaystyle{ \mathbb{R}^n }$ with the Borel sets as measurable subsets) has as probability distribution the measure XP on $\displaystyle{ (\mathcal{X}, \mathcal{A}) }$: the density of $\displaystyle{ X }$ with respect to a reference measure $\displaystyle{ \mu }$ on $\displaystyle{ (\mathcal{X}, \mathcal{A}) }$ is the Radon–Nikodym derivative:

$\displaystyle{ f = \frac{d X_*P}{d \mu} . }$

That is, f is any measurable function with the property that:

That is, f is any measurable function with the property that:

$\displaystyle{ \Pr [X \in A ] = \int_{X^{-1}A} \, d P = \int_A f \, d \mu }$

for any measurable set $\displaystyle{ A \in \mathcal{A}. }$

### Discussion 讨论

In the continuous univariate case above, the reference measure is the Lebesgue measure. The probability mass function of a discrete random variable is the density with respect to the counting measure over the sample space (usually the set of integers, or some subset thereof).

In the continuous univariate case above, the reference measure is the Lebesgue measure. The probability mass function of a discrete random variable is the density with respect to the counting measure over the sample space (usually the set of integers, or some subset thereof).

It is not possible to define a density with reference to an arbitrary measure (e.g. one can't choose the counting measure as a reference for a continuous random variable). Furthermore, when it does exist, the density is almost everywhere unique.

It is not possible to define a density with reference to an arbitrary measure (e.g. one can't choose the counting measure as a reference for a continuous random variable). Furthermore, when it does exist, the density is almost everywhere unique.

## Further details 更多细节

Unlike a probability, a probability density function can take on values greater than one; for example, the uniform distribution on the interval [0, ½] has probability density f(x) = 2 for 0 ≤ x ≤ ½ and f(x) = 0 elsewhere.

Unlike a probability, a probability density function can take on values greater than one; for example, the uniform distribution on the interval [0, ½] has probability density f(x) = 2 for 0 ≤ x ≤ ½ and f(x) = 0 elsewhere.

The standard normal distribution has probability density

The standard normal distribution has probability density

$\displaystyle{ f(x) = \frac{1}{\sqrt{2\pi}}\, e^{-x^2/2}. }$

If a random variable X is given and its distribution admits a probability density function f, then the expected value of X (if the expected value exists) can be calculated as

If a random variable X is given and its distribution admits a probability density function f, then the expected value of X (if the expected value exists) can be calculated as

$\displaystyle{ \operatorname{E}[X] = \int_{-\infty}^\infty x\,f(x)\,dx. }$

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point.

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point.

A distribution has a density function if and only if its cumulative distribution function F(x) is absolutely continuous. In this case: F is almost everywhere differentiable, and its derivative can be used as probability density:

A distribution has a density function if and only if its cumulative distribution function F(x) is absolutely continuous. In this case: F is almost everywhere differentiable, and its derivative can be used as probability density:

$\displaystyle{ \frac{d}{dx}F(x) = f(x). }$

If a probability distribution admits a density, then the probability of every one-point set {a} is zero; the same holds for finite and countable sets.

If a probability distribution admits a density, then the probability of every one-point set {a} is zero; the same holds for finite and countable sets.

Two probability densities f and g represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero.

Two probability densities f and g represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero.

In the field of statistical physics, a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function. This alternate definition is the following:

In the field of statistical physics, a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function. This alternate definition is the following:

If dt is an infinitely small number, the probability that X is included within the interval (tt + dt) is equal to f(tdt, or:

If dt is an infinitely small number, the probability that X is included within the interval (t, t + dt) is equal to f(t) dt, or:

$\displaystyle{ \Pr(t\lt X\lt t+dt) = f(t)\,dt. }$

## Link between discrete and continuous distributions 离散和连续分布之间的联系

It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function, by using the Dirac delta function. (This is not possible with a probability density function in the sense defined above, it may be done with a distribution.) For example, consider a binary discrete random variable having the Rademacher distribution—that is, taking −1 or 1 for values, with probability ½ each. The density of probability associated with this variable is:

It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function, by using the Dirac delta function. (This is not possible with a probability density function in the sense defined above, it may be done with a distribution.) For example, consider a binary discrete random variable having the Rademacher distribution—that is, taking −1 or 1 for values, with probability ½ each. The density of probability associated with this variable is:

$\displaystyle{ f(t) = \frac{1}{2}(\delta(t+1)+\delta(t-1)). }$

More generally, if a discrete variable can take n different values among real numbers, then the associated probability density function is:

More generally, if a discrete variable can take n different values among real numbers, then the associated probability density function is:

$\displaystyle{ f(t) = \sum_{i=1}^np_i\, \delta(t-x_i), }$

where $\displaystyle{ x_1\ldots,x_n }$ are the discrete values accessible to the variable and $\displaystyle{ p_1,\ldots,p_n }$ are the probabilities associated with these values.

where $\displaystyle{ x_1\ldots,x_n }$ are the discrete values accessible to the variable and $\displaystyle{ p_1,\ldots,p_n }$ are the probabilities associated with these values.

This substantially unifies the treatment of discrete and continuous probability distributions. For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability.

This substantially unifies the treatment of discrete and continuous probability distributions. For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability.

## Families of densities 密度族

It is common for probability density functions (and probability mass functions) to be parametrized—that is, to be characterized by unspecified parameters. For example, the normal distribution is parametrized in terms of the mean and the variance, denoted by $\displaystyle{ \mu }$ and $\displaystyle{ \sigma^2 }$ respectively, giving the family of densities.

It is common for probability density functions (and probability mass functions) to be parametrized—that is, to be characterized by unspecified parameters. For example, the normal distribution is parametrized in terms of the mean and the variance, denoted by $\displaystyle{ \mu }$ and $\displaystyle{ \sigma^2 }$ respectively, giving the family of densities.

$\displaystyle{ f(x;\mu,\sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }. }$

It is important to keep in mind the difference between the domain of a family of densities and the parameters of the family. Different values of the parameters describe different distributions of different random variables on the same sample space (the same set of all possible values of the variable); this sample space is the domain of the family of random variables that this family of distributions describes. A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the normalization factor of a distribution (the multiplicative factor that ensures that the area under the density—the probability of something in the domain occurring— equals 1). This normalization factor is outside the kernel of the distribution.

It is important to keep in mind the difference between the domain of a family of densities and the parameters of the family. Different values of the parameters describe different distributions of different random variables on the same sample space (the same set of all possible values of the variable); this sample space is the domain of the family of random variables that this family of distributions describes. A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the normalization factor of a distribution (the multiplicative factor that ensures that the area under the density—the probability of something in the domain occurring— equals 1). This normalization factor is outside the kernel of the distribution.

Since the parameters are constants, reparametrizing a density in terms of different parameters, to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones. Changing the domain of a probability density, however, is trickier and requires more work: see the section below on change of variables.

Since the parameters are constants, reparametrizing a density in terms of different parameters, to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones. Changing the domain of a probability density, however, is trickier and requires more work: see the section below on change of variables.

## Densities associated with multiple variables 与多个变量相关的密度

For continuous random variables X1, ..., Xn, it is also possible to define a probability density function associated to the set as a whole, often called joint probability density function. This density function is defined as a function of the n variables, such that, for any domain D in the n-dimensional space of the values of the variables X1, ..., Xn, the probability that a realisation of the set variables falls inside the domain D is

For continuous random variables X1, ..., Xn, it is also possible to define a probability density function associated to the set as a whole, often called joint probability density function. This density function is defined as a function of the n variables, such that, for any domain D in the n-dimensional space of the values of the variables X1, ..., Xn, the probability that a realisation of the set variables falls inside the domain D is

$\displaystyle{ \Pr \left( X_1,\ldots,X_n \isin D \right) = \int_D f_{X_1,\ldots,X_n}(x_1,\ldots,x_n)\,dx_1 \cdots dx_n. }$

If F(x1, ..., xn) = Pr(X1 ≤ x1, ..., Xn ≤ xn) is the cumulative distribution function of the vector (X1, ..., Xn), then the joint probability density function can be computed as a partial derivative

If F(x1, ..., xn) = Pr(X1 ≤ x1, ..., Xn ≤ xn) is the cumulative distribution function of the vector (X1, ..., Xn), then the joint probability density function can be computed as a partial derivative

$\displaystyle{ f(x) = \left.\frac{\partial^n F}{\partial x_1 \cdots \partial x_n} \right|_x }$

### Marginal densities 边际密度

For i = 1, 2, ...,n, let fXi(xi) be the probability density function associated with variable Xi alone. This is called the marginal density function, and can be deduced from the probability density associated with the random variables X1, ..., Xn by integrating over all values of the other n − 1 variables:

For i = 1, 2, ...,n, let fXi(xi) be the probability density function associated with variable Xi alone. This is called the marginal density function, and can be deduced from the probability density associated with the random variables X1, ..., Xn by integrating over all values of the other n − 1 variables:

$\displaystyle{ f_{X_i}(x_i) = \int f(x_1,\ldots,x_n)\, dx_1 \cdots dx_{i-1}\,dx_{i+1}\cdots dx_n . }$

### Independence 独立性

Continuous random variables X1, ..., Xn admitting a joint density are all independent from each other if and only if

Continuous random variables X1, ..., Xn admitting a joint density are all independent from each other if and only if

$\displaystyle{ f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = f_{X_1}(x_1)\cdots f_{X_n}(x_n). }$

### Corollary 推论

If the joint probability density function of a vector of n random variables can be factored into a product of n functions of one variable

If the joint probability density function of a vector of n random variables can be factored into a product of n functions of one variable

$\displaystyle{ f_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = f_1(x_1)\cdots f_n(x_n), }$

(where each fi is not necessarily a density) then the n variables in the set are all independent from each other, and the marginal probability density function of each of them is given by

(where each fi is not necessarily a density) then the n variables in the set are all independent from each other, and the marginal probability density function of each of them is given by

(其中每个 fi 不一定是密度），那么集合中的n 个变量都是相互独立的，其中每个变量的边际概率密度函数为：

$\displaystyle{ f_{X_i}(x_i) = \frac{f_i(x_i)}{\int f_i(x)\,dx}. }$

### Example 示例

This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. Let us call $\displaystyle{ \vec R }$ a 2-dimensional random vector of coordinates (X, Y): the probability to obtain $\displaystyle{ \vec R }$ in the quarter plane of positive x and y is

This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. Let us call $\displaystyle{ \vec R }$ a 2-dimensional random vector of coordinates (X, Y): the probability to obtain $\displaystyle{ \vec R }$ in the quarter plane of positive x and y is

$\displaystyle{ \lt math display="block"\gt \Pr \left( X \gt 0, Y \gt 0 \right) = \int_0^\infty \int_0^\infty f_{X,Y}(x,y)\,dx\,dy.</math\gt }$

## Function of random variables and change of variables in the probability density function 随机变量的函数和概率密度函数中的变量变化

If the probability density function of a random variable (or vector) X is given as fX(x), it is possible (but often not necessary; see below) to calculate the probability density function of some variable Y = g(X). This is also called a “change of variable” and is in practice used to generate a random variable of arbitrary shape fg(X) = fY using a known (for instance, uniform) random number generator.

If the probability density function of a random variable (or vector) X is given as fX(x), it is possible (but often not necessary; see below) to calculate the probability density function of some variable g(X)}}. This is also called a “change of variable” and is in practice used to generate a random variable of arbitrary shape fY}} using a known (for instance, uniform) random number generator.

It is tempting to think that in order to find the expected value E(g(X)), one must first find the probability density fg(X) of the new random variable Y = g(X). However, rather than computing

It is tempting to think that in order to find the expected value E(g(X)), one must first find the probability density fg(X) of the new random variable g(X)}}. However, rather than computing

$\displaystyle{ \operatorname E\big(g(X)\big) = \int_{-\infty}^\infty y f_{g(X)}(y)\,dy, }$

$\displaystyle{ \operatorname E\big(g(X)\big) = \int_{-\infty}^\infty g(x) f_X(x)\,dx. }$

The values of the two integrals are the same in all cases in which both X and g(X) actually have probability density functions. It is not necessary that g be a one-to-one function. In some cases the latter integral is computed much more easily than the former. See Law of the unconscious statistician.

The values of the two integrals are the same in all cases in which both X and g(X) actually have probability density functions. It is not necessary that g be a one-to-one function. In some cases the latter integral is computed much more easily than the former. See Law of the unconscious statistician.

Xg(X) 实际上在所有有概率密度函数的情况下，两个积分的值都是一样的。g 不一定是单射。在某些情况下，后者的积分比前者更容易计算。见无意识统计学家法则（Law of the unconscious statistician）词条。

### Scalar to scalar 标量到标量

Let $\displaystyle{ g:{\mathbb R} \rightarrow {\mathbb R} }$ be a monotonic function, then the resulting density function is

Let $\displaystyle{ g:{\mathbb R} \rightarrow {\mathbb R} }$ be a monotonic function, then the resulting density function is

$\displaystyle{ g:{\mathbb R} \rightarrow {\mathbb R} }$是一个单调的函数，那么得到的密度函数是

$\displaystyle{ f_Y(y) =f_X\big(g^{-1}(y)\big) \left| \frac{d}{dy} \big(g^{-1}(y)\big) \right|. }$

Here g−1 denotes the inverse function.

Here g−1 denotes the inverse function.

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,

$\displaystyle{ \left| f_Y(y)\, dy \right| = \left| f_X(x)\, dx \right|, }$

or

or

$\displaystyle{ f_Y(y) = \left| \frac{dx}{dy} \right| f_X(x) = \left| \frac{d}{dy} (x) \right| f_X(x) = \left| \frac{d}{dy} \big(g^{-1}(y)\big) \right| f_X\big(g^{-1}(y)\big) = {\big|\big(g^{-1}\big)'(y)\big|} \cdot f_X\big(g^{-1}(y)\big) . }$

For functions that are not monotonic, the probability density function for y is

For functions that are not monotonic, the probability density function for y is

$\displaystyle{ \sum_{k=1}^{n(y)} \left| \frac{d}{dy} g^{-1}_{k}(y) \right| \cdot f_X\big(g^{-1}_{k}(y)\big), }$

where n(y) is the number of solutions in x for the equation $\displaystyle{ g(x)=y }$, and $\displaystyle{ g_k^{-1}(y) }$ are these solutions.

where n(y) is the number of solutions in x for the equation $\displaystyle{ g(x)=y }$, and $\displaystyle{ g_k^{-1}(y) }$ are these solutions.

### Vector to vector 向量到向量

The above formulas can be generalized to variables (which we will again call y) depending on more than one other variable. f(x1, ..., xn) shall denote the probability density function of the variables that y depends on, and the dependence shall be y = g(x1, …, xn). Then, the resulting density function is[citation needed]

The above formulas can be generalized to variables (which we will again call y) depending on more than one other variable. f(x1, ..., xn) shall denote the probability density function of the variables that y depends on, and the dependence shall be g(x1, …, xn)}}. Then, the resulting density function is

$\displaystyle{ \int\limits_{y = g(x_1, \ldots, x_n)} \frac{f(x_1,\ldots, x_n)}{\sqrt{\sum_{j=1}^n \frac{\partial g}{\partial x_j}(x_1, \ldots, x_n)^2}} \,dV, }$

where the integral is over the entire (n − 1)-dimensional solution of the subscripted equation and the symbolic dV must be replaced by a parametrization of this solution for a particular calculation; the variables x1, ..., xn are then of course functions of this parametrization.

where the integral is over the entire (n − 1)-dimensional solution of the subscripted equation and the symbolic dV must be replaced by a parametrization of this solution for a particular calculation; the variables x1, ..., xn are then of course functions of this parametrization.

This derives from the following, perhaps more intuitive representation: Suppose x is an n-dimensional random variable with joint density f. If y = H(x), where H is a bijective, differentiable function, then y has density g:

This derives from the following, perhaps more intuitive representation: Suppose x is an n-dimensional random variable with joint density f. If H(x)}}, where H is a bijective, differentiable function, then y has density g:

$\displaystyle{ g(\mathbf{y}) = f\Big(H^{-1}(\mathbf{y})\Big)\left\vert \det\left[\frac{dH^{-1}(\mathbf{z})}{d\mathbf{z}}\Bigg \vert_{\mathbf{z}=\mathbf{y}}\right]\right \vert }$

with the differential regarded as the Jacobian of the inverse of H(.), evaluated at y.[5]

with the differential regarded as the Jacobian of the inverse of H(.), evaluated at y.

For example, in the 2-dimensional case x = (x1x2), suppose the transform H is given as y1 = H1(x1x2), y2 = H2(x1x2) with inverses x1 = H1−1(y1y2), x2 = H2−1(y1y2). The joint distribution for y = (y1, y2) has density[6]

For example, in the 2-dimensional case x = (x1, x2), suppose the transform H is given as y1 = H1(x1, x2), y2 = H2(x1, x2) with inverses x1 = H1−1(y1, y2), x2 = H2−1(y1, y2). The joint distribution for y = (y1, y2) has density

$\displaystyle{ g(y_1,y_2) = f_{X_1,X_2}\big(H_1^{-1}(y_1,y_2), H_2^{-1}(y_1,y_2)\big) \left\vert \frac{\partial H_1^{-1}}{\partial y_1} \frac{\partial H_2^{-1}}{\partial y_2} - \frac{\partial H_1^{-1}}{\partial y_2} \frac{\partial H_2^{-1}}{\partial y_1} \right\vert. }$

### Vector to scalar 向量到标量

Let $\displaystyle{ V:{\mathbb R}^n \rightarrow {\mathbb R} }$ be a differentiable function and $\displaystyle{ X }$ be a random vector taking values in $\displaystyle{ {\mathbb R}^n }$, $\displaystyle{ f_X(\cdot) }$ be the probability density function of $\displaystyle{ X }$ and $\displaystyle{ \delta(\cdot) }$ be the Dirac delta function. It is possible to use the formulas above to determine $\displaystyle{ f_Y(\cdot) }$, the probability density function of $\displaystyle{ Y=V(X) }$, which will be given by

Let $\displaystyle{ V:{\mathbb R}^n \rightarrow {\mathbb R} }$ be a differentiable function and $\displaystyle{ X }$ be a random vector taking values in $\displaystyle{ {\mathbb R}^n }$, $\displaystyle{ f_X(\cdot) }$ be the probability density function of $\displaystyle{ X }$ and $\displaystyle{ \delta(\cdot) }$ be the Dirac delta function. It is possible to use the formulas above to determine $\displaystyle{ f_Y(\cdot) }$, the probability density function of $\displaystyle{ Y=V(X) }$, which will be given by

$\displaystyle{ V:{\mathbb R}^n \rightarrow {\mathbb R} }$ 是一个可微函数，$\displaystyle{ X }$ 是一个在$\displaystyle{ {\mathbb R}^n }$中取值的随机向量，$\displaystyle{ f_X(\cdot) }$$\displaystyle{ X }$的概率密度函数，$\displaystyle{ \delta(\cdot) }$ 是狄拉克δ函数（Dirac delta Function）。可以使用上述公式来确定$\displaystyle{ f_Y(\cdot) }$，即$\displaystyle{ Y=V(X) }$的概率密度函数，它将由以下公式给出：

$\displaystyle{ f_Y(y) = \int_{{\mathbb R}^n} f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \,d \mathbf{x}. }$

This result leads to the Law of the unconscious statistician:

This result leads to the Law of the unconscious statistician:

$\displaystyle{ \operatorname{E}_Y[Y]=\int_{{\mathbb R}} y f_Y(y) dy = \int_{{\mathbb R}} y \int_{{\mathbb R}^n} f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \,d \mathbf{x} dy = \int_{{\mathbb R}^n} \int_{{\mathbb R}} y f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \, dy d \mathbf{x}= \int_{{\mathbb R}^n} V(\mathbf{x}) f_{X}(\mathbf{x}) d \mathbf{x}=\operatorname{E}_X[V(X)]. }$

Proof:

Proof:

Let $\displaystyle{ Z }$ be a collapsed random variable with probability density function $\displaystyle{ p_Z(z)=\delta(z) }$ (i.e. a constant equal to zero). Let the random vector $\displaystyle{ \tilde{X} }$ and the transform $\displaystyle{ H }$ be defined as

Let $\displaystyle{ Z }$ be a collapsed random variable with probability density function $\displaystyle{ p_Z(z)=\delta(z) }$ (i.e. a constant equal to zero). Let the random vector $\displaystyle{ \tilde{X} }$ and the transform $\displaystyle{ H }$ be defined as

$\displaystyle{ Z }$是一个坍缩的随机变量（collapsed random variable），其概率密度函数$\displaystyle{ p_Z(z)=delta(z) }$（即一个等于0的常数）。设随机向量$\displaystyle{ \tilde{X} }$和变换$\displaystyle{ H }$定义为：

$\displaystyle{ H(Z,X)=\begin{bmatrix} Z+V(X)\\ X\end{bmatrix}=\begin{bmatrix} Y\\ \tilde{X}\end{bmatrix} }$.

It is clear that $\displaystyle{ H }$ is a bijective mapping, and the Jacobian of $\displaystyle{ H^{-1} }$ is given by:

It is clear that $\displaystyle{ H }$ is a bijective mapping, and the Jacobian of $\displaystyle{ H^{-1} }$ is given by:

$\displaystyle{ \frac{dH^{-1}(y,\tilde{\mathbf{x}})}{dy\,d\tilde{\mathbf{x}}}=\begin{bmatrix} 1 & -\frac{dV(\tilde{\mathbf{x}})}{d\tilde{\mathbf{x}}}\\ \mathbf{0}_{n\times1} & \mathbf{I}_{n\times n} \end{bmatrix} }$,

which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that

which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that

$\displaystyle{ f_{Y,X}(y,x) = f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) }$,

which if marginalized over $\displaystyle{ x }$ leads to the desired probability density function.

which if marginalized over $\displaystyle{ x }$ leads to the desired probability density function.

## Sums of independent random variables 独立随机变量之和

The probability density function of the sum of two independent random variables U and V, each of which has a probability density function, is the convolution of their separate density functions:

The probability density function of the sum of two independent random variables U and V, each of which has a probability density function, is the convolution of their separate density functions:

$\displaystyle{ f_{U+V}(x) = \int_{-\infty}^\infty f_U(y) f_V(x - y)\,dy = \left( f_{U} * f_{V} \right) (x) }$

It is possible to generalize the previous relation to a sum of N independent random variables, with densities U1, ..., UN:

It is possible to generalize the previous relation to a sum of N independent random variables, with densities U1, ..., UN:

$\displaystyle{ f_{U_1 + \cdots + U_N}(x) = \left( f_{U_1} * \cdots * f_{U_N} \right) (x) }$

This can be derived from a two-way change of variables involving Y=U+V and Z=V, similarly to the example below for the quotient of independent random variables.

This can be derived from a two-way change of variables involving Y=U+V and Z=V, similarly to the example below for the quotient of independent random variables.

## Products and quotients of independent random variables

Given two independent random variables U and V, each of which has a probability density function, the density of the product Y = UV and quotient Y=U/V can be computed by a change of variables.

Given two independent random variables U and V, each of which has a probability density function, the density of the product Y = UV and quotient Y=U/V can be computed by a change of variables.

### Example: Quotient distribution 示例：商的分布

To compute the quotient Y = U/V of two independent random variables U and V, define the following transformation:

To compute the quotient Y = U/V of two independent random variables U and V, define the following transformation:

$\displaystyle{ Y=U/V }$

$\displaystyle{ Z=V }$

Then, the joint density p(y,z) can be computed by a change of variables from U,V to Y,Z, and Y can be derived by marginalizing out Z from the joint density.

Then, the joint density p(y,z) can be computed by a change of variables from U,V to Y,Z, and Y can be derived by marginalizing out Z from the joint density.

The inverse transformation is

The inverse transformation is

$\displaystyle{ U = YZ }$

$\displaystyle{ V = Z }$

The Jacobian matrix $\displaystyle{ J(U,V\mid Y,Z) }$ of this transformation is

The Jacobian matrix $\displaystyle{ J(U,V\mid Y,Z) }$ of this transformation is

$\displaystyle{ \left| \det\begin{bmatrix} \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \end{bmatrix} \right| = \left| \det\begin{bmatrix} z & y \\ 0 & 1 \end{bmatrix} \right| = |z| . }$

Thus:

Thus:

$\displaystyle{ p(y,z) = p(u,v)\,J(u,v\mid y,z) = p(u)\,p(v)\,J(u,v\mid y,z) = p_U(yz)\,p_V(z)\, |z| . }$

And the distribution of Y can be computed by marginalizing out Z:

And the distribution of Y can be computed by marginalizing out Z:

Y的分布可以通过边际化Z 来计算:

$\displaystyle{ p(y) = \int_{-\infty}^\infty p_U(yz)\,p_V(z)\, |z| \, dz }$

This method crucially requires that the transformation from U,V to Y,Z be bijective. The above transformation meets this because Z can be mapped directly back to V, and for a given V the quotient U/V is monotonic. This is similarly the case for the sum U + V, difference U − V and product UV.

This method crucially requires that the transformation from U,V to Y,Z be bijective. The above transformation meets this because Z can be mapped directly back to V, and for a given V the quotient U/V is monotonic. This is similarly the case for the sum U + V, difference U − V and product UV.

Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables.

Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables.

### Example: Quotient of two standard normals 示例：两个标准正态（变量）的商

Given two standard normal variables U and V, the quotient can be computed as follows. First, the variables have the following density functions:

Given two standard normal variables U and V, the quotient can be computed as follows. First, the variables have the following density functions:

$\displaystyle{ p(u) = \frac{1}{\sqrt{2\pi}} e^{-\frac{u^2}{2}} }$

$\displaystyle{ p(v) = \frac{1}{\sqrt{2\pi}} e^{-\frac{v^2}{2}} }$

We transform as described above:

We transform as described above:

$\displaystyle{ Y=U/V }$

$\displaystyle{ Z=V }$

\displaystyle{ \begin{align} p(y) &= \int_{-\infty}^\infty p_U(yz)\,p_V(z)\, |z| \, dz \\[5pt] &= \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} y^2 z^2} \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} z^2} |z| \, dz \\[5pt] &= \int_{-\infty}^\infty \frac{1}{2\pi} e^{-\frac{1}{2}\left(y^2+1\right)z^2} |z| \, dz \\[5pt] &= 2\int_0^\infty \frac{1}{2\pi} e^{-\frac{1}{2}\left(y^2+1\right)z^2} z \, dz \\[5pt] &= \int_0^\infty \frac{1}{\pi} e^{-\left(y^2+1\right)u} \, du && u=\tfrac{1}{2}z^2\\[5pt] &= \left. -\frac{1}{\pi \left(y^2+1\right)} e^{-\left(y^2+1\right)u}\right|_{u=0}^\infty \\[5pt] &= \frac{1}{\pi \left(y^2+1\right)} \end{align} }

This is the density of a standard Cauchy distribution.

This is the density of a standard Cauchy distribution.

## References 参考

1. Grinstead, Charles M.; Snell, J. Laurie (2009). "Conditional Probability - Discrete Conditional". Grinstead & Snell's Introduction to Probability. Orange Grove Texts. ISBN 161610046X. Retrieved 2019-07-25.
2. Probability distribution function PlanetMath -{zh-cn:互联网档案馆; zh-tw:網際網路檔案館; zh-hk:互聯網檔案館;}-存檔，存档日期2011-08-07.
3. Ord, J.K. (1972) Families of Frequency Distributions, Griffin. (for example, Table 5.1 and Example 5.4)
4. Devore, Jay L.; Berk, Kenneth N. (2007). Modern Mathematical Statistics with Applications. Cengage. p. 263. ISBN 0-534-40473-1.
5. David, Stirzaker (2007-01-01). Elementary Probability. Cambridge University Press. ISBN 0521534283. OCLC 851313783.

## Bibliography 引用

• Pierre Simon de Laplace

1812年). Analytical Theory of Probability.

| title = Analytical Theory of Probability}}


| title = 分析概率理论}

The first major treatise blending calculus with probability theory, originally in French: Théorie Analytique des Probabilités.
The first major treatise blending calculus with probability theory, originally in French: Théorie Analytique des Probabilités.


• Andrei Nikolajevich Kolmogorov (1950
| title = Foundations of the Theory of Probability| url = https://archive.org/details/foundationsofthe00kolm}}


The modern measure-theoretic foundation of probability theory; the original German version (Grundbegriffe der Wahrscheinlichkeitsrechnung) appeared in 1933.
The modern measure-theoretic foundation of probability theory; the original German version (Grundbegriffe der Wahrscheinlichkeitsrechnung) appeared in 1933.


• Patrick Billingsley

1979年). 概率与测量. New York, Toronto, London: John Wiley and Sons

| isbn = 0-471-00710-2}}


| isbn = 0-471-00710-2}}

• David Stirzaker

}}

}}

Chapters 7 to 9 are about continuous variables.
Chapters 7 to 9 are about continuous variables.


}}

}}

• Weisstein, Eric W. "概率密度函数". MathWorld.

Category:Functions related to probability distributions

Category:Concepts in physics

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