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De…nition 1
A
binomial experiment
is an experiment which satis…es each of the following conditions:
1.
The experiment consists of a sequence of
n
trials, where
n
is …xed in advance of the experiment.
2.
The trials are identical, and each trial can result in one of the same two possible outcomes, which
we denote by success (
S
) or failure (
F
).
3.
The trials are independent, so that the outcome on any particular trial does not in‡uence the
outcome on any other trial.
4.
The probability of success is constant from trial to trial; we denote this probability by
p
.
Suppose each trial of an experiment can result in a success or failure, but the sampling is done
without replacement from a population of size
N
. If the sample size (number of trials)
n
is
at most 5% of the population size, the experiment can be analyzed as though it were exactly
a binomial experiment.
De…nition 2
Given a binomial experiment consisting of
n
trials, the
binomial random variable
X
associated with this experiment is de…ned as:
X
=
the number of
S
’s among the
n
trials.
Theorem 1
Let
X
be a binomial random variable with parameters
n
and
p
. Then
b
(
x
;
n;p
)
, the
probability mass function of
X
, is given by:
b
(
x
;
n;p
) =
8
<
:
μ
n
x
¶
p
x
(1
¡
p
)
n
¡
x
x
= 0
;
1
;
2
;:::;n
0
otherwise
9
=
;
:
Proof
Observe that the number of ways that in
n
Bernoulli trials
x
(
x
= 0
;
1
;
2
;:::;n
) successes
can occur is equal to the number of di¤erent sequences of length
n
with
x
successes (
S
’s)
and
(
n
¡
x
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This note was uploaded on 01/08/2012 for the course EXST 4050 taught by Professor Staff during the Fall '10 term at LSU.
 Fall '10
 Staff

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