91Ó°ÊÓ

The uniform distribution of follow X on the interval [a, b]. We have to find the mean of X.

For a continuous random variable X, the mean is called the expectation of X and is calculated by

\(E(X) = \int_{ - \infty }^\infty {xf(x)dx} \)where f(x) is the probability density function of X.

For uniform distribution,

\(\begin{array}{l}f(x) = \frac{1}{{b - a}},{\rm{if }}\,{\rm{a < x < b}}\\{\rm{ = 0, otherwise}}\end{array}\)

\(\begin{array}{l}E(X) = \int_{ - \infty }^\infty {xf(x)dx} \\\,\,\,\,\,\,\,\,\,\,\,\,\, = \int_a^b {x\,\frac{1}{{b - a}}dx} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{{b - a}}\,\left[ {\frac{{{x^2}}}{2}} \right]_a^b\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{b + a}}{2}\end{array}\)

So, the mean of X is \(\frac{{b + a}}{2}\)