Meaning in terms of probability

Question

If $A=[a,b]$ is the support of a density function $f(x)$, then $\int_{A}xf(x)\mathrm{d}x$ is the expected value of $x$. Suppose that $C=[c,d]\subset A$, what is the meaning (if any) of $\int_{C}xf(x)\mathrm{d}x$?

Thank you.

Answer 1

Let $X$ be a random variable equipped with this distribution. Then: $$\int_{C}xf\left(x\right)dx=\mathbb{E}\left(X\mid X\in C\right)\times P\left\{ X\in C\right\} $$

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Working under condition $X\in C$ gives somehow a 'new' random variable. It takes values in $C$ and has density: $$g\left(x\right)=\frac{f\left(x\right)}{P\left\{ X\in C\right\} }$$ Note that here $\int_{C}g\left(x\right)dx=1$.

The connected expectation is: $$\int_{C}xg\left(x\right)dx=\frac{\int_{C}xf\left(x\right)dx}{P\left\{ X\in C\right\} }$$ and is written as: $$\mathbb{E}\left(X\mid X\in C\right)$$