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E[f(X)] - Expectation of function of rand. var.

  1. Mar 20, 2009 #1
    Hi quick question:

    Suppose you have a function of random variables given in the following way

    Z=X if condition A
    Z=Y if condition B

    where both X and Y are random variables, and conditions A & B are disjoint.

    Then would the expectation of Z be

    E[Z]=E[X]*Pr(A)+E[Y]*Pr(B)?

    Thanks in advance.
     
    Last edited: Mar 20, 2009
  2. jcsd
  3. Mar 20, 2009 #2
    No you need independence for that, what you really mean is
    [tex]Z=X\mathbf{1}_A+Y\mathbf{1}_B[/tex]
    where 1 is the indicator function. Now take the expectation
    [tex]\mathbb{E}[Z]=\mathbb{E}[X\mathbf{1}_A]+\mathbb{E}[Y\mathbf{1}_B] [/tex].

    Now you know that [itex]\mathbb{E}[\mathbf{1}_A]=\mathbb{P}(A)[/itex], but to separate the expectations, you need independence between X and A, also between Y and B.
     
  4. Mar 20, 2009 #3
    Thank you Focus for your reply. I see my error.
     
    Last edited: Mar 20, 2009
  5. Mar 20, 2009 #4
    You can use
    [tex]\mathbb{E}[f(X)]=\int_{\mathbb{F}}f(x)F(dx)[/tex]
    where F is the law of X. This may be somewhat abstract so if you are working over the reals and have a pdf f_X then
    [tex]\mathbb{E}[f(X)]=\int_{\mathbb{R}}f(x)f_X(x)dx[/tex].
     
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