# Probability Theory - Expectation Problem

## 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...

LCKurtz
Science Advisor
Homework Helper
Gold Member

## 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...

Yes it does. Your first statement says X and Y take on positive integers.

You have figured out that pX(x) = P(X=x) = 2-x, right? And what is your formula for E(X) for a discrete probability function?

But if it tells me that it only takes on positive integers then technically it can take on infinite positive integers right? And from what i gather you need to know the range of x in order to calculate the expectations. :S

LCKurtz
Science Advisor
Homework Helper
Gold Member
But if it tells me that it only takes on positive integers then technically it can take on infinite positive integers right? And from what i gather you need to know the range of x in order to calculate the expectations. :S

The random variable X can take on values 1,2,3,...with various probabilities. You have already figured out P(X = x) = 2-x

In other words
P(X = 1) = 1/2
P(X = 2) = 1/4
P(X = 3) = 1/8
and so on.

So I will ask you again, what is the formula for E(X) when you know its discrete probability function pX(x)?