## Continuous Uniform Distribution Calculator

Use Continuous Uniform Distribution calculator to find the probability density and cumulative probabilities for continuous Uniform distribution with parameter $a$ and $b$.

The continuous uniform distribution is the simplest probability distribution where all the values belonging to its support have the same probability density. It is also known as rectangular distribution.

Uniform Distribution Calculator
Minimum Value $a$:
Maximum Value $b$
Value of x
Results
Probability density : f(x)
Probability X less than x: P(X < x)
Probability X greater than x: P(X > x)

## Definition of Uniform Distribution

A continuous random variable $X$ is said to have uniform distribution with parameter $\alpha$ and $\beta$ if its p.d.f. is given by

$$f(x; \alpha,\beta) = \dfrac{1}{\beta-\alpha}; \alpha< x< \beta$$

## Distribution function of Uniform Distribution

Distribution function of continuous uniform distribution is

$$F(x) = \dfrac{x-\alpha}{\beta-\alpha}; \alpha< x< \beta$$

## Mean of Uniform Distribution

The mean of uniform distribution is $E(X) = \dfrac{\alpha+\beta}{2}$.

## Variance of Uniform Distribution

The variance of uniform distribution is $V(X) = \dfrac{(\beta - \alpha)^2}{2}$.

## Continous Uniform Distribution Example

The waiting time at a bus stop is uniformly distributed between 1 and 10 minute.

a. What is the probability density function?

b. What is the probability that the rider waits 8 minutes or less?

c. What is the expected waiting time?

d. What is standard deviation of waiting time?

### Solution

Let $X$ denote the waiting time at a bust stop. The waiting time at a bus stop is uniformly distributed between 1 and 10 minute. That is $X\sim U(1,10)$.

a. The probability density function of $X$ is

\begin{aligned} f(x) & = \frac{1}{10-1},\; 1\leq x \leq 10\\ & = \frac{1}{9},\; 1\leq x \leq 10. \end{aligned}

The probability density function : 0.1111

b. The probability that the rider waits 8 minutes or less is

\begin{aligned} P(X\leq 8) & = \int_1^8 f(x) \; dx\\ & = \frac{1}{9}\int_1^8 \; dx\\ & = \frac{1}{9} \big[x \big]_1^8\\ &= \frac{1}{9}\big[ 8-1\big]\\ &= \frac{7}{9}\\ &= 0.7778. \end{aligned}

lower cumulative distribution : 0.7778

c. The expected wait time is $E(X) =\dfrac{\alpha+\beta}{2} =\dfrac{1+10}{2} = 5.5$

d. The variance of waiting time is $V(X) =\dfrac{(\beta-\alpha)^2}{10} =\dfrac{(10-1)^2}{10} = 8.1$.

Hope you like Continous Uniform Distribution Calculator. Click on Examples button to understand several examples solved using uniform distribution.