Which of the following are characteristics of a discrete uniform distribution?

What is a discrete uniform distribution?

Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)

It’s when all the distinct random variables have the exact same probability values, so everything is constant or just a number.

Let’s walk through has this works!

In a uniform probability distribution, all random variables have the same or uniform probability; thus, it is referred to as a discrete uniform distribution.

Imagine a box of 12 donuts sitting on the table, and you are asked to randomly select one donut without looking. Each of the 12 donuts has an equal chance of being selected. Therefore, the probability of any one donut being chosen is the same or uniform.

In fact, we can represent this idea using a simple graph as follows.

Uniform Distribution Graph

But notice that we can show this graphical representation as a density curve of a uniform distribution as a set of rectangles all having equal heights.

PMF Graph Of A Discrete Uniform Distribution

And, what’s important to note is that the value of the total area under any density curve equals one. Therefore, for a discrete uniform distribution, the probability mass function is

PMF Of A Discrete Uniform Random Variable

Moreover, if X is a uniform random variable for a is less than or equal to b, then the values of the mean and variance of a discrete uniform distribution is seen below. In addition, some additional derivations may be seen by Steven Wilson on MileFoot.

Mean And Variance Of Uniform Distribution

So, using our previous example of the box of 12 donuts, where you randomly select one donut without looking. Let’s identify the distribution and calculate it’s mean and variance.

Example Of Discrete Uniform Distribution

In this video, we will use the properties of discrete uniform distributions to identify the probability mass function along with the mean and variance.

Discrete Uniform Distribution – Lesson & Examples (Video)

21 min

  • Introduction to Video: Discrete Uniform Distributions
  • 00:00:36 – How to create, identify and graph a discrete uniform distribution? (Examples #1-2)
  • Exclusive Content for Members Only
  • 00:05:59 – Formulas for finding the mean and variance of a discrete uniform distribution (Example #3)
  • 00:11:44 – Write the uniform distribution and find the mean and variance (Example #4)
  • 00:14:13 – Find the mean and variance given the range of a distinct uniform random variable (Example #5)
  • 00:15:59 – Find the expected value and variance of X for the random variable (Example #6a)
  • 00:17:31 – Determine the mean and variance after the transformation of uniform random variable (Example #6b)
  • Practice Problems with Step-by-Step Solutions
  • Chapter Tests with Video Solutions

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What Is Uniform Distribution?

In statistics, uniform distribution refers to a type of probability distribution in which all outcomes are equally likely. A deck of cards has within it uniform distributions because the likelihood of drawing a heart, a club, a diamond, or a spade is equally likely. A coin also has a uniform distribution because the probability of getting either heads or tails in a coin toss is the same.

The uniform distribution can be visualized as a straight horizontal line, so for a coin flip returning a head or tail, both have a probability p = 0.50 and would be depicted by a line from the y-axis at 0.50.

Key Takeaways

  • Uniform distributions are probability distributions with equally likely outcomes.
  • In a discrete uniform distribution, outcomes are discrete and have the same probability.
  • In a continuous uniform distribution, outcomes are continuous and infinite.
  • In a normal distribution, data around the mean occur more frequently.
  • The frequency of occurrence decreases the farther you are from the mean in a normal distribution.

Understanding Uniform Distribution

There are two types of uniform distributions: discrete and continuous. The possible results of rolling a die provide an example of a discrete uniform distribution: it is possible to roll a 1, 2, 3, 4, 5, or 6, but it is not possible to roll a 2.3, 4.7, or 5.5. Therefore, the roll of a die generates a discrete distribution with p = 1/6 for each outcome. There are only 6 possible values to return and nothing in between.

The plotted results from rolling a single die will be discretely uniform, whereas the plotted results (averages) from rolling two or more dice will be normally distributed.

Some uniform distributions are continuous rather than discrete. An idealized random number generator would be considered a continuous uniform distribution. With this type of distribution, every point in the continuous range between 0.0 and 1.0 has an equal opportunity of appearing, yet there is an infinite number of points between 0.0 and 1.0.

There are several other important continuous distributions, such as the normal distribution, chi-square, and Student's t-distribution.

There are also several data generating or data analyzing functions associated with distributions to help understand the variables and their variance within a data set. These functions include probability density function, cumulative density, and moment generating functions.

Visualizing Uniform Distributions

A distribution is a simple way to visualize a set of data. It can be shown either as a graph or in a list, revealing which values of a random variable have lower or higher chances of happening. There are many different types of probability distributions, and the uniform distribution is perhaps the simplest of them all. 

Under a uniform distribution, each value in the set of possible values has the same possibility of happening. When displayed as a bar or line graph, this distribution has the same height for each potential outcome. In this way, it can look like a rectangle and therefore is sometimes described as the rectangular distribution. If you think about the possibility of drawing a particular suit from a deck of playing cards, there is a random yet equal chance of pulling a heart as there is for pulling a spade—that is, 1/4 or 25%.

The roll of a single dice yields one of six numbers: 1, 2, 3, 4, 5, or 6. Because there are only 6 possible outcomes, the probability of you landing on any one of them is 16.67% (1/6). When plotted on a graph, the distribution is represented as a horizontal line, with each possible outcome captured on the x-axis, at the fixed point of probability along the y-axis.

Uniform Distribution of one six-sided die. Image by Julie Bang © Investopedia 2020

Uniform Distribution vs. Normal Distribution

Probability distributions help you decide the probability of a future event. Some of the most common probability distributions are discrete uniform, binomial, continuous uniform, normal, and exponential. Perhaps one of the most familiar and widely used is the normal distribution, often depicted as a bell curve.

Normal distributions show how continuous data is distributed and assert that most of the data is concentrated on the mean or average. In a normal distribution, the area under the curve equals 1 and 68.27% of all data falls within 1 standard deviation—how dispersed the numbers are—from the mean; 95.45% of all data falls within 2 standard deviations from the mean, and approximately 99.73% of all data falls within 3 standard deviations from the mean. As the data moves away from the mean, the frequency of data occurring decreases.

Discrete uniform distribution shows that variables in a range have the same probability of occurring. There are no variations in probable outcomes and the data is discrete, rather than continuous. Its shape resembles a rectangle, rather than the normal distribution's bell. Like a normal distribution, however, the area under the graph is equal to 1.

Example of Uniform Distribution

There are 52 cards in a traditional deck of cards. In it are four suits: hearts, diamonds, clubs, and spades. Each suit contains an A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, and 2 jokers. However, we'll do away with the jokers and face cards for this example, focusing only on number cards replicated in each suit. As a result, we are left with 40 cards, a set of discrete data.

Suppose you want to know the probability of pulling a 2 of hearts from the modified deck. The probability of pulling a 2 of hearts is 1/40 or 2.5%. Each card is unique; therefore, the likelihood that you will pull any one of the cards in the deck is the same.

Now, let's consider the likelihood of pulling a heart from the deck. The probability is significantly higher. Why? We are now only concerned with the suits in the deck. Since there are only four suits, pulling a heart yields a probability of 1/4 or 25%.

Uniform Distribution FAQs

What Does Uniform Distribution Mean?

Uniform distribution is a probability distribution that asserts that the outcomes for a discrete set of data have the same probability.

What Is the Formula for Uniform Distribution?

The formula for a discrete uniform distribution is

P x = 1 n where: P x = Probability of a discrete value n = Number of values in the range \begin{aligned}&P_x = \frac{ 1 }{ n } \\&\textbf{where:} \\&P_x = \text{Probability of a discrete value} \\&n = \text{Number of values in the range} \\\end{aligned} Px=n1where:Px=Probability of a discrete valuen=Number of values in the range

As with the example of the die, each side contains a unique whole number. The probability of rolling the die and getting any one number is 1/6, or 16.67%.

Is a Uniform Distribution Normal?

Normal indicates the way data is distributed about the mean. Normal data shows that the probability of a variable occurring around the mean, or the center, is higher. Fewer data points are observed the farther you move away from this average, meaning the probability of a variable occurring far away from the mean is lower. The probability is not uniform with normal data, whereas it is constant with a uniform distribution. Therefore, a uniform distribution is not normal.

What Is the Expectation of a Uniform Distribution?

It is expected that a uniform distribution will result in all possible outcomes having the same probability. The probability for one variable is the same for another.

What are the characteristics of a discrete uniform distribution?

1. Discrete uniform distribution. In statistics and probability theory, a discrete uniform distribution is a statistical distribution where the probability of outcomes is equally likely and with finite values. A good example of a discrete uniform distribution would be the possible outcomes of rolling a 6-sided die.

Which of the following are characteristics of the binomial distribution?

The binomial distribution describes the behavior of a count variable X if the following conditions apply: 1: The number of observations n is fixed. 2: Each observation is independent. 3: Each observation represents one of two outcomes ("success" or "failure").

What is the expectation of discrete uniform distribution?

Let X be a discrete random variable with the discrete uniform distribution with parameter n. Then the expectation of X is given by: E(X)=n+12.

Which of the following is variance of discrete uniform distribution?

The mean of the discrete uniform distribution with parameter N is (N + 1)/2. The variance is (N2 – 1)/12.

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