Question

Problem 4 Suppose X and Y have joint PDF Ixr(zy)-{0,y, otherwise, o< <p (a) Find E[XY] (b) Find E[X] (c) Find the Covariance of X and Y

Answer 1

Given,

$\small f_{X,Y}(x,y) = 1/y, 0<y<1, 0<x<y$

(a)

E(XY)

$\small E(XY) =\int \int xy f_{X,Y}(x,y)dxdy$

$\small E(XY) = \int_{0}^{1} \left [\int_{0}^{y}(xy\times (1/y) dx \right ] dy$

$\small = \int_{0}^{1}\left [\frac{x^{2}}{2}\right ]_{x=0}^{y} dy = \int_{0}^{1}\left [\frac{y^{2}}{2}\right ] dy=\left [ \frac{y^{3}}{3\times 2} \right ]_{y=0}^{1} = \frac{1}{6}$

E(XY) = 1/6

(b)

$\small E(X) =\int \int x f_{X,Y}(x,y)dxdy$

$\small E(X) =\int_{0}^{1} \left [\int_{0}^{y}(x\times (1/y) dx \right ] dy$

$\small = \int_{0}^{1}\left [\frac{x^{2}}{2y}\right ]_{x=0}^{y} dy = \int_{0}^{1}\left [\frac{y^{2}}{2y}\right ] dy=\int_{0}^{1}\left [\frac{y}{2}\right ] dy=\left [ \frac{y^{2}}{4} \right ]_{y=0}^{1} = \frac{1}{4}$

E(X) = 1/4

(c)

Cov(X,Y) = E(XY) - E(X) E(Y)

$\small E(Y) =\int \int y f_{X,Y}(x,y)dxdy$

$\small E(Y) =\int_{0}^{1} \left [\int_{0}^{y}(y\times (1/y) dx \right ] dy$

$\small = \int_{0}^{1}\left [x\right ]_{x=0}^{y} dy = \int_{0}^{1}\left[ y \right ] dy=\left [ \frac{y^{2}}{2} \right ]_{y=0}^{1} = \frac{1}{2}$

E(Y) = 1/2

Cov(X,Y) = E(XY) - E(X) E(Y) = 1/6 - (1/4)*(1/2) = 1/6 - 1/8 = 1/24

Cov(X,Y) = 1/24