Question

consider two random variables x and y with joint pmf given by pxy(k,l)=12k l,for k,l=1,2,3,... show that x and y are independent and find the marginal pmfs of x and y. find p(x2 y2≤10)

Answer 1

Explanation:

To show that X and Y are independent, we need to check that their joint PMF factorizes into the product of their marginal PMFs, i.e., PXY(k,l) = PX(k)PY(l) for all k,l.

To do this, we need to find the marginal PMFs of X and Y. We can do this by summing over all possible values of the other variable, as follows:

PX(k) = ∑l=1,2,3,... PXY(k,l) = ∑l=1,2,3,... 1/(2^(k+l))