Formula for the expected value of a function

[gsingh2011]

The formula for the expected value of a continuous random variable is $E(x) = \int_{-\infty}^{\infty} x\cdot f(x)$. This leads me to believe that the expected value of a function g(x) is $E(x) = \int_{-\infty}^{\infty} g(x)\cdot f(g(x))$. However, the correct formula is $E(x) = \int_{-\infty}^{\infty} g(x)\cdot f(x)$. Why is this?

 

[Stephen Tashi]

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.