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 Math Help > Statistics > Binomial Distribution

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Some Discrete Distributions
• The Binomial Distribution
• ## The Binomial Distribution

Perhaps the most commonly used discrete probability distribution is the binomial distribution. An experiment which follows a binomial distribution will satisfy the following requirements (think of repeatedly flipping a coin as you read these):

1. The experiment consists of n identical trials, where n is fixed in advance.
2. Each trial has two possible outcomes, S or F, which we denote ``success'' and ``failure'' and code as 1 and 0, respectively.
3. The trials are independent, so the outcome of one trial has no effect on the outcome of another.
4. The probability of success, , is constant from one trial to another.

The random variable X of a binomial distribution counts the number of successes in n trials. The probability that X is a certain value x is given by the formula

where Recall that the quantity , ``n choose x,'' above is

where

We could use the formulas previously given to compute the mean and variance of X. However, for the binomial distribution these will always be equal to

NOTE:\ A random variable X which follows the binomial distribution can be abbreviated , where is read ``is distributed as.''

NOTE:\ A particularly important example of the use of the binomial distribution is when sampling with replacement (this implies that is constant).

EXAMPLE:\ Suppose we have 10 balls in a bowl, 3 of the balls are red and 7 of them are blue. Define success S as drawing a red ball. If we sample with replacement, P(S)=0.3 for every trial. Let's say n=20, then and we can figure out any probability we want. For example,

The mean and variance are

• The Hypergeometric Distribution
• ## The Hypergeometric Distribution

EXAMPLE:\ Assume we are playing a card game with a regular deck of 52 cards, where 16 of these are ``face cards'' and each ``hand'' consists of 10 randomly selected cards. Using combinations, find the probability of getting 4 face cards in a hand of 10 cards. We know there are possible ``hands'' and possible hands with 4 face cards, therefore

The distribution we derived above is called the hypergeometric distribution. If we can define objects as success/failure, and we have N total items (N=52 cards), with M successes in the population (M=16 face cards) and n items chosen (n=10 cards per hand), then

For our previous example,

For the hypergeometric distribution,

If we let , then

Note that except for , this is the same as the mean and the variance for the binomial distribution. C is called the finite population correction.

• The Negative Binomial Distribution
• ## The Negative Binomial Distribution

The negative binomial distribution is used when the number of successes is fixed and we're interested in the number of failures before reaching the fixed number of successes. An experiment which follows a negative binomial distribution will satisfy the following requirements:

1. The experiment consists of a sequence of independent trials.
2. Each trial has two possible outcomes, S or F.
3. The probability of success, , is constant from one trial to another.
4. The experiment continues until a total of r successes are observed, where r is fixed in advance.
A random variable X which follows a negative binomial distribution is denoted . Its probabilities are computed with the formula

for Formulas for E(X) and for the negative binomial distribution are given by

EXAMPLE:\ Suppose we are at a rifle range with an old gun that misfires 5 out of 6 times. Define ``success'' as the event the gun fires and let X be the number of failures before the third success. Then . The probability that there are 10 failures before the third success is given by

The expected value and variance of X are

• The Poisson Distribution
• ## The Poisson Distribution

The Poisson distribution is most commonly used to model the number of random occurrences of some phenomenon in a specified unit of space or time. For example,

• The number of phone calls received by a telephone operator in a 10-minute period.
• The number of flaws in a bolt of fabric.
• The number of typos per page made by a secretary.
For a Poisson random variable, the probability that X is some value x is given by the formula

where is the average number of occurrences in the specified interval. For the Poisson distribution,

EXAMPLE:\ The number of false fire alarms in a suburb of Houston averages 2.1 per day. Assuming that a Poisson distribution is appropriate, the probability that 4 false alarms will occur on a given day is given by

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• Computer Lab
• # Computer Lab

Applicable StataQuest Commands:

Statistics Correlation Pearson (or Spearman)

Statistics Simple regression to obtain the coefficients for the least square (regression) line

Graphs Scatterplots Plot Y vs. X, with regression line

• Concept Lab
• # Concept Lab

• Ch 3: Relative Frequency and Probability
• Ch 2: Random Sampling

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The webmaster and author of this Math Help site is Graeme McRae.