8.5: Chapter Review

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Binomial Distribution

A statistical experiment can be classified as a binomial experiment if the following conditions are met:

There are a fixed number of trials, \(n\).
There are only two possible outcomes, called "success" and, "failure" for each trial. The letter \(p\) denotes the probability of a success on one trial and \(q\) denotes the probability of a failure on one trial.
The \(n\) trials are independent and are repeated using identical conditions.

The outcomes of a binomial experiment fit a binomial probability distribution. The random variable \(X=\) the number of successes obtained in the \(n\) independent trials. The mean of \(X\) can be calculated using the formula \(\mu=n p\), and the standard deviation is given by the formula \(\sigma=\sqrt{n p q}\).

Geometric Distribution

There are three characteristics of a geometric experiment:

There are one or more Bernoulli trials with all failures except the last one, which is a success.
In theory, the number of trials could go on forever. There must be at least one trial.
The probability, \(p\), of a success and the probability, \(q\), of a failure are the same for each trial.

In a geometric experiment, define the discrete random variable \(X\) as the number of independent trials until the first success. We say that X has a geometric distribution and write \(X \sim G(p)\) where \(p\) is the probability of success in a single trial.

The mean of the geometric distribution \(X \sim G(p)\) is \(\mu=\frac{1}{p}\) and the standard deviation is \(\sigma \sqrt{\frac{(1-p)}{p^2}}=\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}\).

Poisson Distribution

A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is "small" (less than or equal to 0.05) and the number of trials is "large" (greater than or equal to 20).

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Binomial Distribution

Geometric Distribution

Poisson Distribution