# Discrete Uniform Random Variables, a Bernoulli Random Variables, and Binomial Random Variables

Probability distributions have different shapes and characteristics. As such, we describe a random variable based on the shape of the underlying distribution.

## A Discrete Uniform Random Variable

A discrete uniform random variable is one whose probabilities for all possible outcomes are equal. For example, if we throw a die, the probability of any value between 1 and 6 is 1/6. Therefore, the throw of a die is a uniform distribution with a discrete random variable.

In finance, uniform discrete random variables are usually used in simulations. In such instances, financial managers might be interested in drawing a random number such that each random number within a given range has the same probability of being selected.

## A Bernoulli Random Variable

A Bernoulli trial is an experiment that has only two outcomes: success (S) or failure (F). You should note that we use the words “success” and “failure” just for labeling purposes. These words, therefore, may not necessarily carry with them their denotative meanings.

We define the random variable X by X(s) = 1, X(f) = 0. As such, X is the number of successes that occur (0 or 1).

A Bernoulli variable can sometimes be used as an “indicator” to indicate the occurrence of a given event. We could set X = 1 if event B occurs and X = 0 if event B does not occur. For example, event B could be a return of over 10% on a stock.

## A Binomial Random Variable

A binomial random variable is the number of successes in n Bernoulli trials where:

1. the trials are independent – the outcome of any trial does not depend on the outcomes of the other trials;
2. the trials are identical (the probability of success is equal for all trials).

For example, the tossing of a coin has two mutually exclusive outcomes, where the probability of the outcome of any toss (trial) is not affected by outcomes of prior trials.

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