Expectation of 2 random variables

[kingwinner]

Let X and Y be two random variables.

Say, for example, they have the following joint probability mass function

              x
        -1   0    1
  -1    0   1/4  0
y 0     1/4  0  1/4
   1     0   1/4  0

What is the proper way of computing E(XY)?

Can I let Z=XY and find E(Z)=∑ z P(Z=z) ? Would this give E(XY)?

Thanks for explaining!

 

[Hurkyl]

You could do that. But it's more straightforward to simply average over all of the events in your probability space rather than doing a transformation like that.

 

[kingwinner]

But it's more straightforward to simply average over all of the events in your probability space.

How?

 

[HallsofIvy]

What is the probability that xy is 1? In order that xy= 1, either x= 1 and y= 1, which has probability 0 or x= -1 and y= -1 which has probability 0: The probability that xy= 1 is 0.

What is the probability that xy= 0? In order that xy= 0, either x= 0 and y= -1, which has probability 1/4, or x= 0 and y= 0, which has probability 0, or x= 0 and y= 1 which has probability 1/4, or x= -1 and y= 0 which has probability 1/4, or x= 1 and y= 0 which has probability 1/4. The probability that xy= 0 is 1/4+ 1/4+ 1/4+ 1/4= 1.

What is the probability that x= -1? In order that xy= 1, either x= 1 and y= -1 which has probability 0 or x= -1 and y= 1 which has probability 0. The probability that xy= -1 is 0.

The expected value of xy is (-1)(0)+ (0)(1)+ (1)(0)= 0.

Of course, the fact that xy had to be 0 was obvious from the start!

 

[kingwinner]

Yes, this is pretty much the way I was thinking about: Let Z=XY, and find E(Z)=∑ z P(Z=z)

But I also saw a theorem:

E(XY)=∑ ∑ xy P(X=x and Y=y)
      x y

What does the double sum mean? Does it just mean summing over all possible combinations of x and y?

 

[HallsofIvy]

Yes, this is pretty much the way I was thinking about: Let Z=XY, and find E(Z)=∑ z P(Z=z)

But I also saw a theorem:

E(XY)=∑ ∑ xy P(X=x and Y=y)
      x y

What does the double sum mean? Does it just mean summing over all possible combinations of x and y?

Yes. that is exactly what it means.

Strictly speaking what it means is "first sum over all values of y, keeping x as a "variable", then sum that over all values of x" but the effect is to sum over all combinations of x and y.