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Proof of the total probability rule for expected value?

  1. Feb 1, 2017 #1
    1. The problem statement, all variables and given/known data
    Does anyone know of a simple proof for this: https://s30.postimg.org/tw9cjym9t/expect.png

    E(X) = E(X|S)P(S) + E(X|S_c)P(S_c)

    X is a random variable,
    S is an a scenario that affects the likelihood of X. So P(S) is the probability of the scenario occurring and and P(S_c) is the probability of the scenario not occurring

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Feb 1, 2017 #2

    PeroK

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    How do you define ##E(X)##?
     
  4. Feb 1, 2017 #3
    the expected value of the random variable X; the probability weighted average of the possible outcomes of X
     
  5. Feb 1, 2017 #4

    PeroK

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    You can't prove anything with just words. You need a mathematical definition.
     
  6. Feb 1, 2017 #5
    Do you mean this:
    ##E(X) = \sum_{i} X_i P(X_i)##
     
  7. Feb 1, 2017 #6

    PeroK

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    If you have some properties for ##P## you could take it from there.
     
  8. Feb 1, 2017 #7
    Not sure about a mathematical proof, but doesn't that formula just state the obvious? Perhaps putting it into words makes it clearer.

    The expected value of X is the sum of the expected value of X when S happens multiplied by the probability that S happens plus the expected value of X when S doesn't happen times the probability of S not happening.

    Because S happening and S not happening are mutually exclusive you can just add the two values together.

    For a mathematical proof, you'd probably want to include your definition of E(X), the fact that P(S) + P(S') = 1, and the basic conditional probability formula (https://en.wikipedia.org/wiki/Conditional_probability)

    Then go from there.
     
  9. Feb 1, 2017 #8

    Ray Vickson

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    You need formulas for ##E(X|S)## and ##E(X|S_c)##. Do you know what they are?
     
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