What are discrete probability distributions?

Discrete probability distribution

Discrete probability distribution approximates the probabilities of random variables with discrete outcomes.

A discrete random variable has countable values, such as a list of non-negative integers. Probability Mass Functions (PMF) computes the probability of discrete random variables.

For example, if you're counting the number of balls in a box in 10 seconds, you can count 10 or 12 balls but nothing in between.

Types

A discrete probability distribution is divided into various types. Some of the widely used ones are elucidated below.

Binomial distribution

Binomial distribution is generated when there are just two possible outcomes for a random variable. It is obtained by repeating the experiment and calculating the likelihood each time.

For example, a binomial distribution predicts the probability of success (head) when the coin is tossed $n$ times in the air.

The following function is used for calculating binomial distribution:

Where ${n \choose x}$ denotes the number of combinations, $p$ represents the likelihood that an event will succeed, $q$ represents the probability that the event would fail, $n$ is the number of trials, and $x$ is the number of times n trials for a certain outcome.

Multinomial distribution

Multinomial distribution extends the binomial distribution to $k$ categories rather than binary (success/failure). It provides the probability of any particular number of successes for the various categories.

For example, multinomial distribution evaluates the combined probability that team A will win $x1$ times, team B will win $x2$ times, and team C will win $xk$ times.

The probability mass function is given as follows:

Where $n$ is the number of trials, $p_1,……p_k$ are the probabilities of the outcomes $x_1……x_k$ respectively.

Bernoulli distribution

Bernoulli distribution performs a single experiment with only two possible outcomes—success or failure. A single success/failure trial is known as Bernoulli trial/experiment, and it serves as the foundation for many of the distributions detailed below. The sequence of outcomes is known as Bernoulli process.

Let $p$ represents the likelihood of success, and $1 - p$ represents the likelihood of failure.

For example, a coin is tossed in the air, and the probability of getting the head is $0.6$, so using the Bernoulli distribution, the possibility of getting tails will be $1-0.6=0.4$

Negative binomial distribution

Negative binomial distribution is a type of binomial distribution with an unfixed number of trials, $n$, and a random variable $Y$ equals the number of attempts required to get $r$ successes. The Pascal distribution refers to negative binomial distribution.

For example, shuffle a standard deck of cards and pick a card. Repeat until you have drawn three kings. Y is the number of draws required to get three kings. Because the number of trials is not fixed, the distribution is negative binomial.

It is defined as follows:

Where $p$ is the success probability, $k$ is the number of observed failures, and $r$ is the intended number of hits until the trial is terminated.

Poisson distribution

Poisson distribution represents the independent events that occur over a particular period.

For example, a website receives 20 visitors per hour on average. The Poisson distribution determines the likelihood that the website receives more than a specific number of visitors in a given hour.

It is represented as follows:

Where $λ$ is the average number of times the event has occurred during a specific period, $x$ is the desired outcome, and $e$ is Euler's numberEuler's number is a constant equal to 2.71828... ..

Geometric distribution

Geometric distribution determines the number of trials required to get the first success in repeated independent Bernoulli trials. It can also be described as the number of failures before the first success.

For example, when a coin is tossed, getting heads on top is regarded as a success, while getting tails on top is considered a failure. The geometric distribution quickly models the likelihood of the number of times a coin must be tossed to get heads on top.

The probability mass function is given as follows:

Where $p$ is the likelihood of hit and $k$ is the number of misses.

Hypergeometric distribution

Hypergeometric distribution refers to the number of hits in a sequence of $k$ draws without replacement from a sample size. The binomial distribution defines the number of successes for draws with replacement.

For example, drawing a black marble is a successful event, whereas not drawing one is a failure. However, each time a marble is drawn, it is not returned to the box, which affects the likelihood of the next try.

The function is given as follows:

Where $k$ is the number of possible victories, $x$ is the desired number of victories, $N$ is the sample size, and $n$ is the number of trials.

Conclusion

One of the key concepts in statistics is the discrete probability distribution and its various types. It is mainly used to forecast the future using a sample from a random experiment. Discrete probability distributions are widely used as standard tools in computer science, health care, insurance, engineering, and even social science.