Marginal Probability Mass Functions

[twoski]

Homework Statement

Discrete random variables X and Y , whose values are positive integers, have the joint
probability mass function pXY(x,y) = 2^-x-y. Determine the marginal probability mass
functions pX(x) and pY(y). Are X and Y independent? Determine E[X], E[Y], and E[XY].

Homework Equations

Independence is determined by whether p(x,y) = p(x)p(y) for all x and y.

The Attempt at a Solution

My notes don't have much on the topic of determining marginal PMF's using a JPMF... I was hoping someone could point me in the right direction.

 

[Ray Vickson]

Homework Statement

Discrete random variables X and Y , whose values are positive integers, have the joint
probability mass function pXY(x,y) = 2^-x-y. Determine the marginal probability mass
functions pX(x) and pY(y). Are X and Y independent? Determine E[X], E[Y], and E[XY].

Homework Equations

Independence is determined by whether p(x,y) = p(x)p(y) for all x and y.

The Attempt at a Solution

My notes don't have much on the topic of determining marginal PMF's using a JPMF... I was hoping someone could point me in the right direction.

The marginal pmf of X is
$$p_X(x) = P\{ X = x, Y \leq \infty\} = \sum_{\text{all }y} p(x,y).$$
BTW: it is bad form to use the same symbol p to stand for three different things in the same problem. Instead, use subscripts, like this ($p_X(x), p_Y(y)$) or different letters, like this: $g(x)$ and $h(y)$.

 

[twoski]

The marginal pmf of X is
$$p_X(x) = P\{ X = x, Y \leq \infty\} = \sum_{\text{all }y} p(x,y).$$
BTW: it is bad form to use the same symbol p to stand for three different things in the same problem. Instead, use subscripts, like this ($p_X(x), p_Y(y)$) or different letters, like this: $g(x)$ and $h(y)$.

So if that's the marginal PMF of x, then for y...

$$p_Y(y) = P\{ X \leq \infty\ , Y = y} = \sum_{\text{all }x} p(x,y).$$

 

[Karnage1993]

So if that's the marginal PMF of x, then for y...

$$p_Y(y) = P\{ X \leq \infty\ , Y = y\} = \sum_{\text{all }x} p_{X,Y}(x,y).$$

Fixed it, and yes that is correct.

 

[twoski]

So using these 2 PMFs i have to determine whether X and Y are independent. Going by the definition i'd say they are independent.

Is this right?

$$E[X] = \sum_{\text{k}} x_k * p_X(x_k) = \sum_{\text{k}} x_k * \sum_{\text{x}} p(x,y).$$

 

[Karnage1993]

Your $E(X)$ is right. For the next question, use the theorem: If $X,Y$ are independent, then $E(XY) = E(X)E(Y)$.

 

[Ray Vickson]

So using these 2 PMFs i have to determine whether X and Y are independent. Going by the definition i'd say they are independent.

Is this right?

$$E[X] = \sum_{\text{k}} x_k * p_X(x_k) = \sum_{\text{k}} x_k * \sum_{\text{x}} p(x,y).$$

Your equation makes no sense: it is essentially summing over x twice, and not doing anything with y.