Uniform distribution is an important part of statistics or we can say it is the simplest statistical distribution. Although there is hardly any variable that follows a uniform probability distribution. Uniform distribution is a type of probability distribution which has all outcomes equally likely. For example an ordinary deck of fine cards.

Basically we deal with two types of uniform distribution as mentioned below:
  • Continuous uniform distribution 
  • Discreet uniform distribution

Examples 

Some solved examples of uniform distribution are give bellow, 
Example 1:

Suppose in a quiz there are $30$ participants. A question is given to all $30$ participants and the time allowed to answer it is $25$ seconds. Find the probability of participants responds within $6$ seconds?

Solution:

Given 

Interval of probability distribution = [$0$ seconds, $25$ seconds]

Density of probability = $\frac{1}{25-0}$$\frac{1}{25}$ 

Interval of probability distribution of successful event = [$0$ seconds, $6$ seconds]

The probability P(x<6) 

The probability ratio = $\frac{6}{25}$

There are $30$ participants in the quiz 

Hence the participants likely to answer it in $6$ seconds = $\frac{6}{25}$ $\times\ 30\ \approx\ 7$
Example 2:

Suppose a flight is about to land and the announcement says that the expected time to land is $30$ minutes. Find the probability of getting flight land between $25$ to $30$ minutes?

Solution: 

Given 

Interval of probability distribution = [$0$ minutes, $30$ minutes]

Density of probability = $\frac{1}{30-0}$ =$\frac{1}{30}$ 

Interval of probability distribution of successful event = [$0$ minutes, $5$ minutes]

The probability (25 < x < 30)

The probability ratio = $\frac{5}{30}$ = $\frac{1}{6}$

Hence the probability of getting flight land between $25$ minutes to $30$ minutes = $0.16$
Example 3:

Suppose a random number $N$ is taken from $690$ to $850$ in uniform distribution. Find the probability number $N$ is greater than the $790$?

Solution:

Given 

Interval of number in probability distribution = $[690 ,\ 850]$

Density of probability = $\frac{1}{850 - 690}$ =$\frac{1}{160}$ 

Now the probability is (790 < x < 850)

Interval of probability distribution of successful event = $[790,\ 850]$ = $60$

The probability ratio = $\frac{60}{160}$ = $\frac{1}{10}$

Hence the probability of $N$ number greater than $790$  = $0.1$
Example 4:

Suppose a train is delayed by approximately $60$ minutes. What is the probability that train will reach by $57$ minutes to $60$ minutes?

Solution:

Given 

Interval of probability distribution = [$0$ minutes, $60$ minutes]

Density of probability = $\frac{1}{60 - 0}$ =$\frac{1}{60}$ 

Interval of probability distribution of successful event = [$0$ minutes, $3$ minutes]

Now the probability is (57 < x < 60)

The probability ratio = $\frac{3}{60}$

Hence the probability of train to reach between $57$ to $60$ minutes = $\frac{3}{60}$ = $0.05$