Example A mobile computer is moving in the region A bounded by the x axis, the line x = 1, and the line y = x. If (X, Y) denotes the position of the computer at a given time, the joint distribution of X and Y is given by f(x, y) = x + cy² Find Cov(X, Y). 0

Example A mobile computer is moving in the region A bounded by the x axis, the line x = 1, and the line y = x. If (X, Y) denotes the position of the computer at a given time, the joint distribution of X and Y is given by f(x, y) = x + cy² Find Cov(X, Y). 0

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Example
A mobile computer is moving in the region A
bounded by the x axis, the line x = 1, and the
line y = x. If (X, Y) denotes the position of the
computer at a given time, the joint distribution
of X and Y is given by
f(x, y) = x + cy²
Find Cov(X, Y).
0<x<1, 0<y<1

Expert Solution

Step 1

The joint distribution is given as follows.

$f (x, y) = x + c y^{2} 0 < x < 1, 0 < y < 1$

The covariance formula is as follows.

$C o v (X, Y) = E [X Y] - E [X] E [Y]$

$\begin{array}{rcl} E [X] & = & \int_{x} x f (x) d x \\ E [Y] & = & \int_{x} y f (y) d y \\ E [X Y] & = & \int_{y} \int_{x} x y f (x, y) d x d y \end{array}$

The marginal p.d.f of $X$ is

$f (x) = \int_{y} f (x, y) . d y$

The marginal p.d.f of $Y$ is

$f (y) = \int_{x} f (x, y) . d x$

For the joint distribution

$\int_{x} \int_{y} f (x, y) d x d y = 1$

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