## 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 | |
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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.