X Uniform(a,b). Find E[X].

Set up the integral. E[X]= ab x 1b-a,dx. Evaluate. The integral evaluates to the midpoint of the interval.

Mean of a Uniform Distribution

Statistics

Continuous Distributions

Easy

X \sim Uniform (a, b)

. Find

E [X]

Hint 1

The uniform density is constant, so the mean coincides with the midpoint of the interval.

Hint 2

Integrate $x$ times the constant density $\frac{1}{b - a}$ over $[a, b]$ to confirm.

Show worked solution

1. Set up the integral
$E [X] = \int_{a}^{b} x \cdot \frac{1}{b - a} d x$ .
2. Evaluate
The integral evaluates to the midpoint of the interval.
$E [X] = \frac{a + b}{2}$

Answer: (a+b)/2