Integration and statistics

Have you ever wondered why you see many integration problems of the form $\int xf(x)dx$ or $\int (x-\text{a number})^2 f(x)dx$ in your textbook?

Those problems are related to expected value and variance, which are applied in statistics.

Prerequisite knowledge:
For a discrete random variable, the expected value is the sum of all xP(x). For a continuous random variable, P(x) is the probability density function, and integration takes the place of addition.

Definitions:
Let X be an absolutely continuous random variable with probability density function $f_X(x)$.

The expected value of X is:
$E(X)=\int_{-\infty}^{\infty}xf_X(x)dx$

It should fulfill absolute integrability, i.e. $\int_{-\infty}^{\infty}|x|f_X(x)dx<\infty$. This ensures that the improper integral $\int_{-\infty}^{\infty}xf_X(x)dx$, a shorthand for $\lim\limits_{t \to -\infty} \int_t^0 xf_X(x)dx+\lim\limits_{t \to \infty} \int_0^t xf_X(x)dx$, is well-defined. When the absolute integrability condition is not satisfied, the expected value of X does not exist.

Let f(x) be a probability density function on the domain [a,b], then the variance of f(x) is $\int_a^b(x-\mu)^2 f(x)dx$.

Examples