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Formula for the expected value of a function

  1. Oct 27, 2012 #1
    The formula for the expected value of a continuous random variable is [itex]E(x) = \int_{-\infty}^{\infty} x\cdot f(x)[/itex]. This leads me to believe that the expected value of a function g(x) is [itex]E(x) = \int_{-\infty}^{\infty} g(x)\cdot f(g(x))[/itex]. However, the correct formula is [itex]E(x) = \int_{-\infty}^{\infty} g(x)\cdot f(x)[/itex]. Why is this?
  2. jcsd
  3. Oct 27, 2012 #2

    Stephen Tashi

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    You're talking about a "function of a random variable" (which is itself a random variable) not an ordinary function. The definition of the expectation of a random variable involves an integrand that is the product of the value of the variable times the probability density of the random variable evaluated at that value. If X is a random varaible with density f(x) and Y = g(X) is a function of the random variable X then it is not generally true that the probability density of Y is f(g(x)). Hence f(g(x)) is not the appropriate term to use in the integrand when computing the expectation of Y.
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