Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Expectation of 2 random variables

  1. Nov 5, 2008 #1
    Let X and Y be two random variables.

    Say, for example, they have the following joint probability mass function
    Code (Text):

                  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!
     
  2. jcsd
  3. Nov 5, 2008 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  4. Nov 6, 2008 #3
    How?
     
  5. Nov 6, 2008 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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!
     
  6. Nov 6, 2008 #5
    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:
    Code (Text):
    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?
     
  7. Nov 6, 2008 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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.
     
    Last edited: Nov 6, 2008
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?