Question

Pair of random variables (X, Y ) has range

{(2^(−k), 2^(−l)|l, k are integers and 1 ≤ k < l}.

The joint probability mass function is given by
pX,Y (2^(−k), 2^−(l)) = C*2^(−k−l)
,for some constant C. Find the value of C. Find the marginal probability mass functions
for both X and Y and evaluate E(X) and E(Y ).

Answer 1

The joint pdf of X and Y is given by:

Now, we consider the limits of k and l:

Thus, the given pdf can be written as :

which is the joint pdf of X and Y in simplified form.

Now, we need to find the value of C. We know that:

Thus, the joint pdf can be written as:

Now, we find the marginal pdf of X:

The marginal pdf of Y is given by:

The expectation of Y is given by:

The expectation of X is given by:

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