A continuous random variable is uniformly distributed on an interval when every subinterval of the same length has the same probability.

For $a<b$, the continuous uniform distribution on $[a,b]$ has constant density

$$f(x)=\frac1{b-a},\qquad a\le x\le b,$$

and density zero outside the interval.

The rectangle under the density has total area

$$(b-a)\frac1{b-a}=1.$$

Probabilities are proportional to length

If $a\le c<d\le b$, then

$$P(c\le X\le d) =\int_c^d\frac1{b-a},dx =\frac{d-c}{b-a}.$$

Thus a subinterval occupying one quarter of the total interval carries one quarter of the probability.

Mean

The density is symmetric about the midpoint, so

$$E[X]=\frac{a+b}{2}.$$

The expected value lies at the geometric center of the interval.

When uniformity is plausible

A uniform model expresses complete symmetry among locations in a bounded interval. It can describe an idealized random position or phase when no part of the interval is favored.

It is not appropriate merely because only a minimum and maximum are known; constant density is a substantive modeling assumption.