How Does Changing Variables Affect the Expected Value in Probability Theory?

[Rasalhague]

Hoel: An Introduction to Mathematical Statistics introduces the following formulas for expectation, where the density is zero outside of the interval [a,b].

$$E\left [ X \right ] = \int_{a}^{b} x f(x) \; dx$$

$$E\left [ g(X) \right ] = \int_{a}^{b} g(x) f(x) \; dx$$

He says, "Let the random variable g(X) be denoted by Y. Then knowing the density f(x) of X it is theoretically possible to to find the density h(x) of Y. The expected value of g(X) is the same as the expected value of Y; therefore if h(y) is available, the latter expected value can be expressed in the form

$$E\left [ Y \right ] = \int_{-\infty}^{\infty} y h(y) \; dy.$$

"By using the change of variable techniques of calculus, it can be shown that this value is the same as the value given by (22) [the 2nd formula I've quoted in this post]."


I've been trying to do this. Let I denote the identity function on $\mathbb{R}$. Let $f_X$ denote the pdf of the distribution induced by a random variable X, and $F_X$ its cdf. I'm guessing that when the expected value of a distribution is expressed like this in terms of a random variable, $E[X]$ is to be understood as $E[P_X]$, and $E[g(X)]$ as $E[P_{g \circ X}]$, where $P_X$ means the distribution induced by the random variable X, given some sample space implicit in the context.

Then expectation is defined by

$$E[P_X]=\int_a^b I \cdot f_X,$$

and we must show that

$$\int_a^b I \cdot f_{g \circ X} = \int_a^b g \cdot f_X,$$

or do the limits need to be changed? Using the chain rule (integration by substitution) and the identity

$$F_{g \circ X}=F_X \circ g,$$

leads me to

$$\int_a^b I \cdot f_{g \circ X} = \int_{g(a)}^{g(b)} I \cdot f_X$$

which looks tantalisingly close, but am I going in the right direction?

 

[spamiam]

This is sometimes called The Law of the Unconscious Statistician, so you might try looking for sources. I'm not sure how to use your approach, so I'll give a slightly different one. In Sheldon Ross's A First Course in Probability, he shows this by first proving the lemma
$$\mathbf{E}[Y] =\int_0^\infty \mathbf{P}\{Y > y \} \, dy - \int_0^\infty \mathbf{P}\{Y < -y \} \, dy$$
for any random variable Y. (This is a pretty straightforward proof: just switch the order of integration using the pdf for Y.) After that, he sets Y = g(X) and switching the order of integration once more, the result falls out. I can go into more detail if you'd like, but I hope this helps!