# 3.7.1. Discrete Uniform Probability Distribution

>>Dr. Kash Barker: One last discrete distribution we should chat about briefly is the DISCRETE UNIFORM DISTRIBUTION. Again, as a distribution of a discrete random

variable, the realizations of a uniformly distributed random variable are integer values.

The term “uniform” refers to the fact that each realization has the same likelihood

of occurring. And we mention that it’s a “discrete uniform” distribution, as there’s

also a continuous uniform distribution that we’ll discuss later on.

The realizations of the uniform distribution are bounded by two integer values which we’ll

call a and b. And unlike the other discrete distributions we’ve seen, the lower bound

“a” need not be zero because we’re not necessarily dealing with counts as we were

with the other distributions. Therefore, if there are n possible realizations of the random

variable, then the likelihood of any realization is 1 divided by n.

An easy example of a random variable that follows a uniform distribution would be random

variable Y defined as the value that we roll in a single roll of a die. There are six possibilities

for the outcome of a single roll of a die: 1, 2, 3, 4, 5, or 6. And, assuming no funny

business with the die, all of those roles are equally likely. The set of realizations

is the set containing integers 1 through 6, including 1 and 6. There are six possible

realizations, therefore the probability of any particular realization is 1 divided by

6. The distribution of the role of a die is probably

what you’d expect it to be. Six realizations, each occurring with the probability 0.0167.

The expected value of a discrete uniform distribution is, not surprisingly, just the average of

the two bounds: (a+b) divided by 2. You can check that at home by multiplying each realization

by the probability of that realization, 1 over n, and summing across realizations. For

the die example, the expected value is 1 + 6 divided by 2, or 3.5. If we repeat the experiment

over and over, what would we expect the random variable to be? 3.5. The variance of a discrete

uniform distribution is b minus a plus 1 quantity squared, minus 1 all over 12. For the die rolling

problem, the variance is 2.917. Therefore, the standard deviation would be 1.708.

To recap, the discrete uniform distribution is good for modeling the probability of random

variables whose realizations all have equal likelihoods of occurring. Its construction

is pretty simple relative to the other discrete distributions we’ve discussed.

## One Reply to “3.7.1. Discrete Uniform Probability Distribution”

Thanks.