Uniform Distribution Calculator

What Is the Uniform Distribution?

A continuous uniform distribution $U(a,b)$ assigns equal probability density to every point in the interval $[a,b]$. Its PDF is constant:

$$f(x) = \frac{1}{b-a} \quad \text{for } a \le x \le b$$

Interval probabilities are proportional to length. Related tools: Normal Distribution Calculator and Probability Calculator.

$$F(x) = \frac{x-a}{b-a}, \quad \mu = \frac{a+b}{2}, \quad \sigma^2 = \frac{(b-a)^2}{12}$$

$$Q(p) = a + p(b-a)$$

For $U(0,10)$, the probability that $X$ falls between 4 and 7 is $(7-4)/(10-0) = 0.3$.

Can a and b be equal?

No. The interval length must be positive, so $b$ must be greater than $a$.

What is the PDF outside [a,b]?

It is zero. All probability mass lies inside the interval.

Is uniform distribution the same as normal distribution?

No. Uniform density is flat. Normal density is bell-shaped and concentrates probability near the mean.

Why is variance divided by 12?

It comes from integrating $(x-\mu)^2$ over a symmetric interval with constant density. The result is $(b-a)^2/12$.