mathematics

For all problems on this page, use the following setup:

Let N be a positive integer random variable with PMF of the form

pN(n)=(1/2)*(n)*2^(-n),n=1,2,….

Once we see the numerical value of N , we then draw a random variable K whose (conditional) PMF is uniform on the set {1,2,…,2n} .

Question 1) Write down an expression for the joint PMF pN,K(n,k) .

For n=1,2,… and k=1,2,…,2n

##Find the marginal PMF pK(k) as a function of k . For simplicity, provide the answer only for the case when k is an even number. (The formula for when k is odd would be slightly different, and you do not need to provide it).
For k=2,4,6,… : Question
Question 2) Pk(K)=??
Question 3)Let A be the event that K is even. Find P(A|N=n) and P(A) .

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  1. Pn,k(n,k)=2^(-n)/4
    Pk(k)=(1/2)^((k/2)+1)

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