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(ELEC210)[2010](s)quiz~1980^_10297.pdf
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1. Lets assume that tow random variables X and Y are generated as follows: cos(2/8)sin(2/8)XrandYr......



Where r is fixed but is a discrete random variable uniformly distributed over the set S={0,1,,7}. .

a) Find the joint pmf of X and Y. Are they independent? Are they uncorrelated?

b) Find the marginal pmf of X and of Y.

c) Find the probability of the following event:A {/2,/2}XrYr...




2. Let X and Y be the joint Gaussian random variables with the following joint pdf:
22,221(2)(,)exp()2(1)21XYxxyyfxy..........
,,11xy.........

a) Prove that X and Y are independent random variables if and only if =0 . .



b) Find P[X*Y<0] if =0.



3Let,,and be independent Gaussian random variable, all with zero-mean and unit-variance. Based on them, we generate three new random variables as follows: 1X
2X
3X
4X

,, and . 112YXX..
223YXX..
334YXX..

a) Find the covariance matrix of . 123(,,)YYYY.


b) Find the joint pdf of Y.

c) Find the joint pdf of and;and ; and and. 1Y
2Y
3Y
1Y
3Y


d) Find a new transformation A such that the random vector consists of independent Gaussian random variables. ZAY.