# Discrete Distribution

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## Definition of '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.

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.

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Copyright © 2004-2023, MyPivots. All rights reserved.