How Does Uniform Distribution Work in Probability?
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: Key Concepts, Formulas & Examples
1. What is a uniform distribution in the context of probability?
A uniform distribution, also known as a rectangular distribution, is a type of probability distribution where every possible outcome within a given range is equally likely. This means that each value in the specified interval has the same probability of occurring. It is one of the simplest probability distributions and can be either continuous (for a range of real numbers) or discrete (for a finite set of outcomes).
2. What are the key formulas for a continuous uniform distribution?
For a continuous uniform distribution defined over an interval from a (minimum value) to b (maximum value), the key formulas are:
- Probability Density Function (PDF): f(x) = 1 / (b - a) for a ≤ x ≤ b, and 0 otherwise. This constant value is why the distribution's graph is a rectangle.
- Mean (Expected Value): E(X) = (a + b) / 2. The mean is the midpoint of the interval.
- Variance: Var(X) = (b - a)2 / 12. The variance measures the spread of the distribution.
3. How does a uniform distribution fundamentally differ from a normal distribution?
The fundamental difference lies in how probability is distributed. In a uniform distribution, the probability is constant for all outcomes within its range, creating a flat, rectangular shape. In contrast, a normal distribution has a bell-shaped curve where the probability is highest at the mean and decreases as you move away from it. While both can be symmetric, the uniform distribution assumes all outcomes are equally likely, whereas the normal distribution assumes outcomes cluster around a central value.
4. In what real-world scenarios is a uniform distribution a suitable model?
A uniform distribution is useful for modelling situations where any outcome in a specific range is considered equally possible. Key examples include:
- Random Number Generation: Computer algorithms often generate random numbers from a uniform distribution (e.g., between 0 and 1).
- Simulation of Waiting Times: If a bus arrives every 20 minutes, a person arriving at a random moment has a uniformly distributed waiting time between 0 and 20 minutes.
- Measurement Rounding Errors: The error made when rounding a number to the nearest integer can be approximated by a uniform distribution.
5. Can a uniform distribution be discrete? Provide a common example.
Yes, a uniform distribution can be discrete. A discrete uniform distribution applies to a finite number of outcomes, where each outcome has the same probability. The most common example is rolling a fair six-sided die. There are six possible outcomes {1, 2, 3, 4, 5, 6}, and the probability of rolling any specific number is exactly 1/6, making it a perfect discrete uniform distribution.
6. Why is the uniform distribution often called the 'rectangular distribution'?
The name 'rectangular distribution' comes directly from the visual representation of its Probability Density Function (PDF). When you plot the PDF of a continuous uniform distribution, it forms a perfect rectangle. The base of the rectangle lies on the x-axis from the minimum value 'a' to the maximum value 'b', and its height is constant at 1/(b-a). This distinct rectangular shape makes the name an intuitive alternative.
7. What are the primary limitations of using a uniform distribution to model real-world phenomena?
The main limitation of the uniform distribution is its strict assumption that all outcomes are equally likely. Many real-world processes do not behave this way. For instance, variables like student exam scores, people's heights, or measurement errors tend to cluster around an average value, making a normal distribution a more appropriate model. The uniform distribution's hard boundaries and constant probability are often too simplistic for complex natural or social phenomena.
