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The Binomial Random Variable
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Concept Version 9
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Binomial Probability Distributions

This chapter explores Bernoulli experiments and the probability distributions of binomial random variables.

Learning Objective

  • Apply Bernoulli distribution in determining success of an experiment


Key Points

    • A Bernoulli (success-failure) experiment is performed nnn times, and the trials are independent.
    • The probability of success on each trial is a constant ppp; the probability of failure is q=1−pq=1-pq=1−p.
    • The random variable XXX counts the number of successes in the nnn trials.

Term

  • Bernoulli Trial

    an experiment whose outcome is random and can be either of two possible outcomes, "success" or "failure"


Example

    • At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A "success" could be defined as an individual who withdrew. The random variable is XXX: the number of students who withdraw from the randomly selected elementary physics class.

Full Text

Many random experiments include counting the number of successes in a series of a fixed number of independently repeated trials, which may result in either success or failure. The distribution of the number of successes is a binomial distribution. It is a discrete probability distribution with two parameters, traditionally indicated by nnn, the number of trials, and ppp, the probability of success. Such a success/failure experiment is also called a Bernoulli experiment, or Bernoulli trial; when n=1n=1n=1, the Bernoulli distribution is a binomial distribution.

Named after Jacob Bernoulli, who studied them extensively in the 1600s, a well known example of such an experiment is the repeated tossing of a coin and counting the number of times "heads" comes up.

In a sequence of Bernoulli trials, we are often interested in the total number of successes and not in the order of their occurrence. If we let the random variable XXX equal the number of observed successes in nnn Bernoulli trials, the possible values of XXX are $0, 1, 2, \dots, n$. If xxx success occur, where $x=0, 1, 2, \dots, n$, then n−xn-xn−x failures occur. The number of ways of selecting xxx positions for the xxx successes in the xxx trials is:

(nx)=n!x!(n−x)!\displaystyle {{n}\choose{x}} = \frac{n!}{x!(n-x)!}(​x​n​​)=​x!(n−x)!​​n!​​

Since the trials are independent and since the probabilities of success and failure on each trial are, respectively, ppp and q=1−pq=1-pq=1−p, the probability of each of these ways is px(1−p)n−xp^x(1-p)^{n-x}p​x​​(1−p)​n−x​​. Thus, the p.d.f. of XXX, say f(x)f(x)f(x), is the sum of the probabilities of these (nxnxnx) mutually exclusive events--that is,

f(x)=(nx)px(1−p)n−xf(x)=(nx)p^x(1-p)^{n-x}f(x)=(nx)p​x​​(1−p)​n−x​​ ,  $x=0, 1, 2, \dots, n$

These probabilities are called binomial probabilities, and the random variable XXX is said to have a binomial distribution.

Wind pollination

These male (a) and female (b) catkins from the goat willow tree (Salix caprea) have structures that are light and feathery to better disperse and catch the wind-blown pollen.

Probability Mass Function

A graph of binomial probability distributions that vary according to their corresponding values for nnn and ppp.

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