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.

• 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 and 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 0 \(\leq\)X\(\leq\)23.

Theoretical Mean Formula

Standard Deviation Formula

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.

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