Discrete Distribution

A discrete distribution is a probability distribution that can take on only a finite or countable number of values. In other words, it is a function that assigns a probability to each of a finite or countable number of possible outcomes.

Discrete distributions are often used to model the outcomes of experiments that can have only a limited number of results. For example, we might use a discrete distribution to model the number of heads that we get when we flip a coin 10 times. In this case, the possible outcomes are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 heads.

The probability mass function (PMF) of a discrete distribution is a function that gives the probability of each possible outcome. For example, the PMF of the binomial distribution is given by:

```
P(X = x) = (n! / x!(n - x)!) * p^x * (1 - p)^(n - x)
```

where n is the number of trials, x is the number of successes, and p is the probability of success on each trial.

The cumulative distribution function (CDF) of a discrete distribution is a function that gives the probability that the random variable X is less than or equal to a given value. For example, the CDF of the binomial distribution is given by:

```
F(x) = P(X <= x) = \sum_{i=0}^x P(X = i)
```

Discrete distributions are often used in statistics and probability theory. They are also used in a variety of applications, such as queuing theory, reliability engineering, and economics.

Here are some additional examples of discrete distributions:

* The binomial distribution is used to model the number of successes in a sequence of independent experiments, each of which has a constant probability of success.
* The Poisson distribution is used to model the number of events that occur in a fixed interval of time or space.
* The geometric distribution is used to model the number of trials until the first success.
* The negative binomial distribution is used to model the number of successes until a specified number of failures.

Discrete distributions are a powerful tool for modeling a variety of random phenomena. They are used in a wide range of applications, from statistics and probability theory to queuing theory, reliability engineering, and economics.