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The Laplace mechanism adds Laplace noise (i.e. noise from the [[Laplace distribution]], which can be expressed by probability density function <math>\text{noise}(y)\propto \exp(-|y|/\lambda)\,\!</math>, which has mean zero and standard deviation <math>\sqrt{2} \lambda\,\!</math>). Now in our case we define the output function of <math>\mathcal{A}\,\!</math> as a real valued function (called as the transcript output by <math>\mathcal{A}\,\!</math>) as <math>\mathcal{T}_{\mathcal{A}}(x)=f(x)+Y\,\!</math> where <math>Y \sim \text{Lap}(\lambda)\,\!\,\!</math> and <math>f\,\!</math> is the original real valued query/function we planned to execute on the database. Now clearly <math>\mathcal{T}_{\mathcal{A}}(x)\,\!</math> can be considered to be a continuous random variable, where
The Laplace mechanism adds Laplace noise (i.e. noise from the [[Laplace distribution]], which can be expressed by probability density function <math>\text{noise}(y)\propto \exp(-|y|/\lambda)\,\!</math>, which has mean zero and standard deviation <math>\sqrt{2} \lambda\,\!</math>). Now in our case we define the output function of <math>\mathcal{A}\,\!</math> as a real valued function (called as the transcript output by <math>\mathcal{A}\,\!</math>) as <math>\mathcal{T}_{\mathcal{A}}(x)=f(x)+Y\,\!</math> where <math>Y \sim \text{Lap}(\lambda)\,\!\,\!</math> and <math>f\,\!</math> is the original real valued query/function we planned to execute on the database. Now clearly <math>\mathcal{T}_{\mathcal{A}}(x)\,\!</math> can be considered to be a continuous random variable, where
<math>\displaystyle{\mathrm{pdf}(\mathcal{T}_{\mathcal{A},D_1}(x)=t)}{\mathrm{pdf}(\mathcal{T}_{\mathcal{A},D_2}(x)=t)}=\displaystyle{\text{noise}(t-f(D_1))}{\text{noise}(t-f(D_2))}\,\!</math>
<nowiki>The Laplace mechanism adds Laplace noise (i.e. noise from the Laplace distribution, which can be expressed by probability density function \text{noise}(y)\propto \exp(-|y|/\lambda)\,\!, which has mean zero and standard deviation \sqrt{2} \lambda\,\!). Now in our case we define the output function of \mathcal{A}\,\! as a real valued function (called as the transcript output by \mathcal{A}\,\!) as \mathcal{T}_{\mathcal{A}}(x)=f(x)+Y\,\! where Y \sim \text{Lap}(\lambda)\,\!\,\! and f\,\! is the original real valued query/function we planned to execute on the database. Now clearly \mathcal{T}_{\mathcal{A}}(x)\,\! can be considered to be a continuous random variable, where</nowiki>
<nowiki>The Laplace mechanism adds Laplace noise (i.e. noise from the Laplace distribution, which can be expressed by probability density function \text{noise}(y)\propto \exp(-|y|/\lambda)\,\!, which has mean zero and standard deviation \sqrt{2} \lambda\,\!). Now in our case we define the output function of \mathcal{A}\,\! as a real valued function (called as the transcript output by \mathcal{A}\,\!) as \mathcal{T}_{\mathcal{A}}(x)=f(x)+Y\,\! where Y \sim \text{Lap}(\lambda)\,\!\,\! and f\,\! is the original real valued query/function we planned to execute on the database. Now clearly \mathcal{T}_{\mathcal{A}}(x)\,\! can be considered to be a continuous random variable, where</nowiki>