Difficult# Improper Integral Defining Expected Value of a PDF

APCALC-GWKVYQ

A nonnegative function $f$ is known as a probability density function if $\int_{-\infty }^{\infty }f(t)dt=1$.

For a given probability density function, the expected value is given by $E(x)=\int_{-\infty }^{\infty }tf(t)dt$.

Consider the probability density function:

$$f(t)=\frac{1}{5}e^{-\frac{t}{5}},t\geq 0$$

$$f(t)=0, t<0$$

Find the expected value for $f(t)$.