# Expected value of bounded function? [closed]

The expected value of a function is

$$E[g(x)] = \int_{-\infty}^{\infty}g(x)f(x)dx.$$

What happens if $g$ is a function such as $g:\mathbb{R}\rightarrow]a,b[$? Does the expected value exist? Should I take the integral just as

$$E[g(x)] = \int_{a}^{b}g(x)f(x)dx?$$

• Are you trying to estimate the expected value within the bounds (ie, as it manifests), or of the latent variable? Aug 22 '15 at 20:18
• No, the second formula is incorrect! If the function $g(x)$ is bounded between $a$ and $b$ for all real $x$, it is true that $a E[X]\leq E[g(X)]\leq b E[X]$. Aug 22 '15 at 21:26
• @gung within the bounds, I guess (I don't actually understand what you mean by latent variable). Aug 22 '15 at 21:31
• The expectation may or may not exist. It depends on the distribution $f$ as well. There is no one rule for existence in this general of a context. You can simulate it to find out and see if the central limit theorem holds up. Be sure to intergrate over the entire sample space though...not just $a$ to $b$. Aug 23 '15 at 2:53
• Could you please explain what "$g:\mathbb{R}\rightarrow]a,b[$" means? The standard mathematical meaning is that $g$ is a fucntion defined on $\mathbb{R}$ with values in the open interval $]a,b[$, but with that interpretation this question is so trivial--as @StijnDeVuyst has pointed out--that we have to assume you mean something else by it. What, exactly?
– whuber
Aug 23 '15 at 22:24