5 Pick a non negative integer N randomly such that Probabili

(5) Pick a non negative integer N randomly, such that Probability [N=k] = e^-1/k!, k = 0,1,2?. Such an N is called a Poisson random variable with parameter 1. Flip a fair coin N times and denote by X and Y the number of heads and tails obtained, respectively. (a) Calculate Probability [X = k, Y = l] for k,l >= 0. (b) Calculate Probability [X=k] and Probability[Y=L]. Hint. Recall that summation n=0 to infinity x^n/n! = e^x. (c) Are the random variables X and Y independent? (d) What if the coin is biased?

Solution

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