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