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## Homework Statement

Discrete random variables X and Y , whose values are positive integers, have the joint probability mass function , (, ) = 2−−. Determine the marginal probability mass functions () and (). Are X and Y independent? Determine [], [ ], and [ ].

## The Attempt at a Solution

y=1ʃ∞ pX(x) = Σ pX,Y(x,y)

= y=1ʃ∞ Σ 2^(-x-y)

= y=1ʃ∞ 2^(-x) Σ 2^(-y)

= 2^-x

x=1ʃ∞ pY(y) = Σ pX,Y(x,y)

= x=1ʃ∞ Σ 2^(-x-y)

= x=1ʃ∞ 2^(-y) Σ 2^(-x)

= 2^-y

Since pX,Y(x, y) = pX(x) * pY(y), X and Y are independent of eachother.

I am stuck figuring out the expectations. Are we to assume that x and y can only take the values 0 and 1? The expectation requires a weighted average of all the possible values of x and y but the problem does not tell us the possible values...