ECE PhD QE CNSIP Jan 2006 Problem1 - Rhea

Question

1 (33 points)

Let $ \mathbf{X} $ and $ \mathbf{Y} $ be two joinly distributed random variables having joint pdf

$ f_{\mathbf{XY}}\left(x,y\right)=\left\{ \begin{array}{lll} 1, & & \text{ for }0\leq x\leq1\text{ and }0\leq y\leq1\\ 0, & & \text{ elsewhere. } \end{array}\right. $

(a)

Are $ \mathbf{X} $ and $ \mathbf{Y} $ statistically independent? Justify your answer.

(b)

Let $ \mathbf{Z} $ be a new random variable defined as $ \mathbf{Z}=\mathbf{X}+\mathbf{Y} $ . Find the cdf of $ \mathbf{Z} $ .

(c)

Find the variance of $ \mathbf{Z} $ .

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Part 2.

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Part 3.

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Part 4.

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