In statistics, the uniform distribution is a type of probability distribution in that all the possible outcomes are equally possible. A deck of cards has uniform distributions within it since the probability of drawing a heart, club, diamond or spade is equally possible. A coin also has a uniform distribution since the probability of getting either the heads or the tails in the coin toss is the same. The uniform distribution can be visualized as the straight horizontal line, hence, for a coin flip returning to a head or a tail, both have a probability p = 0.50 and it would be depicted by the line from the y-axis at 0.50. There are two kinds of uniform distributions namely discrete and continuous. In the former type of distribution, each of the possible outcomes is discrete. In continuous distribution, the outcomes are continuous and infinite. In this lesson, we will learn about what is a uniform distribution, the uniform distribution formula, the mean of uniform distribution, the density of uniform distribution, and look at some uniform distribution examples.

Let us learn what is a probability distribution in detail in this section.

A continuous probability distribution is called the uniform distribution and it is related to the events that are equally possible to occur. It is defined by two different parameters, x and y, where x = the minimum value and y = the maximum value. It is generally represented by u(x,y).

If the probability density function or the probability distribution of the uniform distribution with a continuous random variable X is \[f(b) = \frac{1}{y - x}\], it is denoted by U(x, y) where x and y are the constants in a way that x < a < y. It is written in the following manner:

X ~ U(a, b)

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Now that you know about the uniform distribution, let us look at some of the uniform distribution formulae.

The theoretical mean of the uniform distribution is given by:

\[\mu = \frac{(x + y)}{2}\]

The standard deviation formula of the uniform distribution is given by:

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

Some of the examples of the uniform distribution are given as follows. Take a look at them for a better understanding of the topic.

Example 1

The average weight gained by a person over the winter months is uniformly distributed and ranges from 0 to 30 lbs. Find the probability of a person that he will gain between 10 and 15lbs in the winter months.

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Solution:

First, find the total height of the distribution. The area under the probability distribution is always 1. Since there are 30 units starting from 0 to 30) the height is \[\frac{1}{30}\].

Then find the width of the slice of the distribution. Do this with subtracting the biggest number b from the smallest number a and you will get

b – a = 15 – 10 = 5.

Then multiply the width in Step 2 by the height in Step 1 and you will get

Probability = \[5 \times \frac{1}{30} = \frac{5}{30} = \frac{1}{6}\].

Example 2

Determine P(X ≤ 10) for the above-given question.

Solution:

This question is asking you to find the probability which the random variable X is lesser than 10. In simpler words, you need to determine the probability of the person gaining up to ten pounds.

Find the width of the box first which is b – a = 10 – 0 = 10.

Then multiply the width in Step 1 by the height. You already know that the height is \[\frac{1}{30}\].

Hence, \[10 \times \frac{1}{30} = \frac{10}{30} = \frac{1}{3}\].

FAQ (Frequently Asked Questions)

1. What do you Mean by a Uniform Distribution?

The uniform distribution is said to be a continuous distribution that is bounded on both sides. Its density does not rely on the value of x. It is the special case of the Beta distribution. It is frequently also called the rectangular distribution. Most of the random number generators provide samples from a uniform distribution on (0,1) and convert these samples to the random variates from the other distributions.

The uniform distribution is used in representing the random variable with the constant likelihood of being in a small interval between the min and the max. The uniform distribution is generally used if you want your desired results to range between the two numbers.

f(x) = 1/(max - min)

Here, min = minimum x and max = maximum x

2. What is the Difference Between the Uniform Distribution and the Normal Distribution?

A uniform distribution is the one in which all the values are equally possible within a given range. The normal distribution is the one in which the values cluster around the mean or the average, and the outlying values are impossible.

Both the uniform and the normal distributions are symmetric, which means that the median and the mean are equal and all the values in any given range which is higher than the mean are equally possible as the corresponding range which is lower than the mean. For example, for the normal distribution with the mean 5, the range 8 - 9 is possible equally as the range 1 - 2. Both the ranges are at a distance of 3 - 4 from the mean. This is true irrespective of what the standard deviation is, however, the exact chances tend to depend on the standard deviation.