Uniform Distribution Formula Properties and Solved Examples
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.
Uniform Probability Distribution
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.
Theoretical Mean of Uniform Distribution
The theoretical mean of the uniform distribution is given by:
\[\mu = \frac{(x + y)}{2}\]
Standard Deviation Formula of Uniform Distribution
The standard deviation formula of the uniform distribution is given by:
\[\sigma = \sqrt{\frac{(y - x)^{2}}{12}}\]
Uniform Distribution Examples
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, \[10 \times \frac{1}{30} = \frac{10}{30} = \frac{1}{3}\].
FAQs on Uniform Distribution in Statistics and Probability
1. What is a uniform distribution in statistics?
A uniform distribution is a probability distribution in which all outcomes within a given interval are equally likely. In other words, every value in the range has the same probability of occurring.
- For a discrete uniform distribution, each distinct value has equal probability.
- For a continuous uniform distribution, the probability density is constant over an interval [a, b].
- It is often called a rectangular distribution in the continuous case.
2. What is the formula for the continuous uniform distribution?
The probability density function (PDF) of a continuous uniform distribution on the interval [a, b] is f(x) = 1 / (b − a) for a ≤ x ≤ b.
- f(x) = 0 for x outside [a, b].
- The graph is a horizontal line between a and b.
- The total area under the curve equals 1.
3. What is the mean of a uniform distribution?
The mean of a continuous uniform distribution on [a, b] is (a + b) / 2.
- It represents the midpoint of the interval.
- For example, if a = 2 and b = 8, the mean is (2 + 8)/2 = 5.
4. What is the variance of a uniform distribution?
The variance of a continuous uniform distribution on [a, b] is (b − a)2 / 12.
- The standard deviation is the square root: (b − a) / √12.
- Example: If a = 0 and b = 6, variance = 62/12 = 36/12 = 3.
5. How do you find the probability in a continuous uniform distribution?
To find probability in a continuous uniform distribution, use P(c ≤ X ≤ d) = (d − c) / (b − a) for values within [a, b].
- Step 1: Subtract the lower bound from the upper bound of the interval (d − c).
- Step 2: Divide by the total interval length (b − a).
- Example: If X ~ U(0,10), then P(2 ≤ X ≤ 5) = (5 − 2)/10 = 0.3.
6. What is the difference between discrete and continuous uniform distribution?
The key difference is that a discrete uniform distribution applies to distinct values, while a continuous uniform distribution applies to all real numbers in an interval.
- Discrete: Each value has probability 1/n (e.g., fair dice).
- Continuous: Uses constant density 1/(b − a).
- Discrete probabilities are exact values; continuous probabilities are found over intervals.
7. Can you give an example of a uniform distribution?
A classic example of a uniform distribution is rolling a fair six-sided die.
- Each outcome (1, 2, 3, 4, 5, 6) has probability 1/6.
- This is a discrete uniform distribution.
- Another example: choosing a random number between 0 and 1 follows a continuous uniform distribution U(0,1).
8. What are the properties of a uniform distribution?
The main properties of a uniform distribution are constant probability and equal likelihood across its range.
- The PDF is constant: f(x) = 1/(b − a).
- The mean is (a + b)/2.
- The variance is (b − a)2/12.
- The distribution is symmetric about its mean.
9. Why is the uniform distribution called a rectangular distribution?
The continuous uniform distribution is called a rectangular distribution because its probability density graph forms a rectangle.
- The height of the rectangle is 1/(b − a).
- The width is (b − a).
- The total area equals 1, satisfying probability rules.
10. Where is the uniform distribution used in real life?
The uniform distribution is used when outcomes are assumed to be equally likely within a range.
- Random number generation in simulations.
- Modeling arrival times within fixed intervals.
- Quality control when defects are evenly distributed.
- Game theory and probability experiments.
