# Uniform Distribution

Each element or member of the group ( Continuous uniform distribution, rectangular distribution belonging to a symmetric probability distribution) is equally probable for all intervals which have the same length on the distribution’s support in Statistics.

It is defined by two parameters, x and y, where x = minimum value and y = maximum value.

The uniform distribution is generally denoted by u(x,y).

Uniform distribution can be explained well by knowing the following:

1. Definition
2. Practicing solved examples.
3. It’s important formulas

## Uniform Distribution Definition:

• A continuous probability distribution is a Uniform distribution and is related to the events which are equally likely to occur.

OR

• If the probability density function of a uniform distribution with a continuous random variable X is f(b)=$\frac{1}{y-x}$, then It is denoted by U(x,y), where x and y are constants such that x<a<y.
 It is written as: X $\sim$ U(a,b)

## Uniform Distribution Example:

(Note: Check whether the data is inclusive or exclusive before working out problems with uniform distribution.)

Example: The data in the table below are 55 times a baby yawns, in seconds, of a 9-week-old baby girl.

 10.4 19.6 18.8 13.9 17.8 16.8 21.6 17.9 12.5 11.1 4.9 12.8 14 22.8 20.8 15.9 16.3 13.4 17.1 14.5 19 22.8 1.3 0.7 8.9 11.9 10.9 7.3 5.9 3.7 17.9 19.2 9.8 5.8 6.9 2.6 5.8 21.7 11.8 3.4 2.1 4.5 6.3 10.7 8.9 9.7 9.1 7.7 10.1 3.5 6.9 7.8 11.6 13.8 18.6
• The sample mean = 11.49
• The sample standard deviation = 6.23.

As assumed, the yawn times, in secs, it follows a uniform distribution between 0 to 23 seconds(Inclusive).

So, it is equally likely that any yawning time is from 0 to 23.

• Histograph Type: Empirical Distribution (It matches with theoretical uniform distribution).
• If the length is A, in seconds, of a 9-month-old baby’s yawn.

• The uniform distribution notation for the same is A $\sim$ U(x,y) where x = the lowest value of a and y = the highest value of b.
• f(a) = $\frac{1}{y-x}$, f(a) = the probability density function. For x $\leq$a$\leq$y.

In this example:

• X $\sim$ U(0,23)
• f(a) = $\frac{1}{23-0}$ for For 0 $\leq$X$\leq$23.

### Theoretical Mean Formula

 $\mu$ = $\frac{x+y}{2}$

## Standard Deviation Formula

 $\sigma$ = $\sqrt{\frac{(y-x)^{2}}{12}}$

In this example,

The theoretical mean = $\mu$ = $\frac{x+y}{2}$ $\mu$ = $\frac{0+23}{2}$ = 11.50

standard deviation = $\sqrt{ \frac{(y-x)^{2}}{12}}$

standard deviation = $\sqrt{ \frac{(23-0)^{2}}{12}}$ =6.64 seconds.