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Problem 6 Let X and Y be random variables with joint PDF px;y Let Z X/Y. Find an expression (it involves an integral) of the PDF of Z in terms of the joint PDF of X...

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Problem 6 Let X and Y be random variables with joint PDF px;y Let Z X/Y. Find an expression (it involves an integral) of the PDF of Z in terms of the joint PDF of X and Y.ii_ Determine the PDF of Z if2 + y 0 < E,y < 1 otherwise.Px;x(x,y)For the first part, introduce an auxiliary random variable W = Y; find the joint PDF of Z and W, and then integrate Over W_ Notice that px;y (zw , w) 0 if w > 1/2. Check that by integrating the PDF of Z yOu get 1_

Problem 6 Let X and Y be random variables with joint PDF px;y Let Z X/Y. Find an expression (it involves an integral) of the PDF of Z in terms of the joint PDF of X and Y. ii_ Determine the PDF of Z if 2 + y 0 < E,y < 1 otherwise. Px;x(x,y) For the first part, introduce an auxiliary random variable W = Y; find the joint PDF of Z and W, and then integrate Over W_ Notice that px;y (zw , w) 0 if w > 1/2. Check that by integrating the PDF of Z yOu get 1_



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Let $X, Y,$ and $Z$ have the joint probability density function $f(x, y, z)=\left\{\begin{array}{ll}k x y^{2} z, & 0<x, y<1,0<z<2 \\ 0, & \text { elsewhere }\end{array}\right.$ (a) Find $k$. (b) Find $P\left(X<\frac{1}{4}, Y>\frac{1}{2}, 1<Z<2\right)$.


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