2011/09/13
1
Introduction to Probability
and Statistics
Thirteenth Edition
Chapter 5
Several Useful Discrete
Distributions
Introduction
• Discrete random variables take on only a
finite or countably infinite number of
values.
• Three discrete probability distributions
serve as models for a large number of
practical applications:
The
binomial
random variable
The
Poisson
random variable
The
hypergeometric
random
variable
The Binomial Random Variable
• The
cointossing experiment
is
a simple example of a
binomial
random variable.
Toss a fair
coin
n
= 3 times and record
x
=
number of heads.
x
p(x)
0
1/8
1
3/8
2
3/8
3
1/8
The Binomial Random Variable
• Many situations in real life resemble the
coin toss, but the coin is not necessarily
fair, so that P(H)
1/2.
•
Example:
A geneticist samples
10 people and counts the number
who have a gene linked to
Alzheimer’s disease.
Person
•
Coin:
•
Head:
•
Tail:
•
Number of
tosses:
•
P(H):
Has gene
Doesn’t have
gene
n
= 10
P(has gene) =
proportion in the
population who have
the gene.
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The Binomial Experiment
1.
The experiment consists of
n
identical
trials.
2.
Each trial results in
one of two
outcomes
, success (S) or failure (F).
3.
The probability of success
on a single
trial is
p
and
remains constant
from
trial to trial. The probability of failure is
q
= 1 – p.
4.
The trials are
independent
.
5.
We are interested in
x
, the number of
successes in
n
trials.
Binomial or Not?
• Very few real life applications
satisfy these requirements exactly.
• Select two people from the U.S.
population, and suppose that 15% of the
population has the Alzheimer’s gene.
• For the first person,
p
= P(gene) = .15
• For the second person,
p
P(gene) =
.15, even though one person has been
removed from the population.
The Binomial Probability
Distribution
• For a binomial experiment with
n
trials and
probability
p
of success on a given trial,
the probability of
k
successes in
n
trials is
.
1
!
0
1
)
2
)...(
2
)(
1
(
!
)!
(
!
!
.
,...
2
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 Fall '08
 Any
 Statistics, Normal Distribution, Probability, Probability distribution, Probability theory, Binomial distribution, Discrete probability distribution

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