# Uniform Distribution

## What is Uniform Distribution

A continuous probability distribution is a Uniform distribution and is related to the events which are equally likely to occur. It is defined by two parameters, x and y, where x = minimum value and y = maximum value. It is generally denoted by u(x,y).

OR

If the probability density function or probability distribution of a uniform distribution with a continuous random variable X is f(b)=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)

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

## Uniform Distribution Examples

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) = 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) = 1/(23-0) for For 0 $\leq$X$\leq$23.

### Theoretical Mean Formula

 $\mu$ = (x+y)/2

### Standard Deviation Formula

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

In this example,

The theoretical mean = $\mu$ = (x+y)/2

$\mu$ = (0+23)/2 = 11.50

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

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