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Bernoulli Trials and Related Distributions
A single Bernoulli trial is an experiment
with two possible outcomes S and F
such that P(S) = p and P(F) = 1 – p = q.
In practice, the parameter p is often unknown.
Note: Use of p instead of a Greek letter is a violation of the usual convention. Some books use π, but that can lead to confusion with π = 3.14159.
If we have several Bernoulli trials in an experiment,
we will assume they are independent and
that p is the same on each trial.
Bernoulli trials are the basis of three families of distributions:
Bernoulli, Binomial, Geometric, and Negative binomial.
A Bernoulli distribution is simply based on one trial:
P( X = 0) = q , P( X = 1) = p. E( X ) = 0( q ) + 1( p ) = p.
Now we look at Binomial Distributions.
Binomial Experiment.
Consists of a known number n of Bernoulli trials.
The random variable of interest is X = # of Ss in n trials.
Explicitly, the rules for a binomial experiment are:
- Trials independent.
- Two possible outcomes S and F on each trial.
- P(S) = p is the same on each trial.
- Known number n of trials.
- The binomial random variable is X = # of Ss.
Rules 1-3 simply describe repeated Bernoulli trials. Rules 4 & 5 describe how Bernoulli trials are used in a binomial experiment.
The parameters of a binomial experiment are n and p.
In statistical practice, often n is known and p is not.
Both parameters must be known in order to find the
distribution of X.
A Bernoulli distribution is a special case of a binomial distribution in which n = 1.
Here is are the graphs for the distribution that arises from tossing a biased coin with p = 1/3.
Notice that the table in the back of WMS does not give this distribution.
Show that the mean of this distribution BINOM(5, 1/3) is 1.6667.
Examples of distributions that are not binomial. Each of the rules for a binomial experiment is important. Different families of distributions result when rules fail.
A. Suppose there is no fixed n. We toss a fair coin until we see the first Head. X = the number of the trial on which we see the first Head. This is an example of the Geometric Distribution.
B. Suppose there is no fixed n. We toss a fair coin until we see the 3rd Head. X = the number of the trial on which we see the 3rd Head. This is an example of the Negative Binomial Distribution.
C. An urn contains 8 balls, 4 marked S and 4 marked F. We withdraw 5 balls without replacement. The draws are not Bernoulli trials because they are dependent. On each draw the P(S) = 1/2. (Unconditionally on the past.) X = # of Ss drawn. But the distribution of X is not BINOM(5, 1/2) In particular, P( X = 0) = P( X = 5) = 0. This is an example of the Hypergeometric Distribution.
D. A process begins in the AM with P(S) = .99. But during the day P(S) gradually shifts to .95. We sample n = 5 items during the day. The number of Ss observed is not binomial because the p is not the same on all trials. (This is a messy situation, there is no "named" distribution to model it.) Copyright © 2004 by Bruce E. Trumbo. All rights reserved. Department of Statistics, CSU Hayward.WMS=Waclerly, et al.: Mathematical Statistics with Applications, 6th ed., Duxbury, 2002.
