$$X sim extGa(alpha,eta)quad quad quad quadY|X sim extN Big( 0,frac1X Big).$$

How execute I compute the PDF, mean and also variance of the random variable $Z = X imes Y$?




You are watching: Expectation of product of dependent random variables

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The moments are derived making use of iterated moment formulae. Using the legislation of iterated expectation you have:

$$eginequation eginalignedmathbbE(Z) &= mathbbE(mathbbE(Z|X)) \<6pt>&= mathbbE(mathbbE(X cdot Y|X)) \<6pt>&= mathbbE(X cdot mathbbE(Y|X)) \<6pt>&= mathbbE(X cdot 0) \<6pt>&= mathbbE(0) = 0. \<6pt>endaligned endequation$$

Using the law of iterated variance you have:

$$eginequation eginalignedmathbbV(Z) &= mathbbV(mathbbE(Z|X)) + mathbbE(mathbbV(Z|X)) \<6pt>&= mathbbV(mathbbE(X cdot Y|X)) + mathbbE(mathbbV(X cdot Y|X)) \<6pt>&= mathbbV(X cdot mathbbE(Y|X)) + mathbbE(X^2 cdot mathbbV(Y|X)) \<6pt>&= mathbbV(X cdot 0) + mathbbE Big( X^2 cdot frac1X Big) \<6pt>&= mathbbV(0) + mathbbE(X) \<6pt>&= fracalphaeta. \<6pt>endaligned endequation$$

To find the distribution attribute you deserve to use the law of full probcapacity to get:

$$eginequation eginalignedF_Z(z) = mathbbP(Z leqslant z) &= mathbbP(X cdot Y leqslant z) \<6pt>&= int limits_0^infty mathbbP(X cdot Y leqslant z|X=x) cdot f_X(x) dx \<6pt>&= int limits_0^infty Phi Big( fraczsqrtx Big) cdot extGa(x|alpha, eta) dx. \<6pt>endaligned endequation$$

The density corresponding to this circulation feature is:

$$eginequation eginalignedf_Z(z) = fracdF_Zdz(z) &= fracddz int limits_0^infty Phi Big( fraczsqrtx Big) cdot extGa(x|alpha, eta) dx \<6pt>&= int limits_0^infty fracpartialpartial z Phi Big( fraczsqrtx Big) cdot extGa(x|alpha, eta) dx \<6pt>&= int limits_0^infty frac1sqrtx cdot phi Big( fraczsqrtx Big) cdot extGa(x|alpha, eta) dx \<6pt>&= int limits_0^infty x^-1/2 cdot frac1sqrt2 pi exp Big( -frac12 fracz^2x Big) cdot fraceta^alphaGamma(alpha) x^alpha-1 exp(- eta x) dx \<6pt>&= frac1sqrt2 pi fraceta^alphaGamma(alpha) int limits_0^infty x^alpha -3/2 exp Big( - eta x -frac12 fracz^2x Big) dx.

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\<6pt>endaligned endequation$$

Tbelow is no closed-form expression for this integral, and so the thickness cannot be streamlined any type of better. The density deserve to be computed numerically utilizing conventional methods of numerical computation of integrals.