Download discrete probability distribution

CHAPTER 5
Discrete Probability
Distributions
© Copyright McGraw-Hill 2004
5-1
Objectives

Construct a probability distribution for a
random variable.

Find the mean, variance, and expected value
for a discrete random variable.

Find the exact probability for X successes in n
trials of a binomial experiment.
© Copyright McGraw-Hill 2004
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Objectives (cont’d.)

Find the mean, variance, and standard
deviation for the variable of a binomial
distribution.

Find probabilities for outcomes of variables
using the Poisson, hypergeometric, and
multinomial distributions.
© Copyright McGraw-Hill 2004
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Introduction

Many decisions in business, insurance, and
other real-life situations are made by
assigning probabilities to all possible
outcomes pertaining to the situation and
then evaluating the results.
© Copyright McGraw-Hill 2004
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Introduction (cont’d.)

This chapter explains the concepts and
applications of probability distributions. In
addition, special probability distributions,
such as the binomial, multinomial, Poisson,
and hypergeometric distributions are
explained.
© Copyright McGraw-Hill 2004
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Random Variables

A random variable is a variable whose values
are determined by chance.
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Discrete Probability Distribution

A discrete probability distribution consists of
the values a random variable can assume and
the corresponding probabilities of the values.
The probabilities are determined theoretically
or by observation.
© Copyright McGraw-Hill 2004
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Calculating the Mean

In order to find the mean for a probability
distribution, one must multiply each possible
outcome by its corresponding probability and
find the sum of the products.
  X 1  P( X 1 )  X 2  P( X 2 )  X 3  P( X 3 )  . . .  X n  P( X n )
© Copyright McGraw-Hill 2004
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Rounding Rule

The mean, variance, and standard deviation
should be rounded to one more decimal place
than the outcome, X.
© Copyright McGraw-Hill 2004
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Variance of a Probability Distribution

The variance of a probability distribution is
found by multiplying the square of each
outcome by its corresponding probability,
summing those products, and subtracting the
square of the mean.

The formula for calculating the variance is:
 2  [ X 2  P( X )]   2

The formula for the standard deviation is:
 
2
© Copyright McGraw-Hill 2004
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Expected Value

Expected value or expectation is used in
various types of games of chance, in
insurance, and in other areas, such as
decision theory.
© Copyright McGraw-Hill 2004
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Expected Value (cont’d.)

The expected value of a discrete random
variable of a probability distribution is the
theoretical average of the variable. The
formula is:
  EX    X  P X 

The symbol E(X) is used for the expected
value.
© Copyright McGraw-Hill 2004
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The Binomial Distribution

Many types of probability
problems have only two
possible outcomes or they
can be reduced to two
outcomes.

Examples include: when a
coin is tossed it can land
on heads or tails, when a
baby is born it is either a
boy or girl, etc.
© Copyright McGraw-Hill 2004
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The Binomial Experiment

The binomial experiment is a probability
experiment that satisfies these requirements:
1. Each trial can have only two possible
outcomes—success or failure.
2. There must be a fixed number of trials.
3. The outcomes of each trial must be
independent of each other.
4. The probability of success must remain
the same for each trial.
© Copyright McGraw-Hill 2004
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The Binomial Experiment (cont’d.)

The outcomes of a binomial experiment and
the corresponding probabilities of these
outcomes are called a binomial distribution.
© Copyright McGraw-Hill 2004
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Notation for the Binomial Distribution
P( S )
P( F )
p

The symbol for the probability of success

The symbol for the probability of failure

The numerical probability of success
q

The numerical probability of failure
P( S )  p
and
n
The number of trials
X

P (F )  1  p  q
The number of successes
© Copyright McGraw-Hill 2004
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Binomial Probability Formula

In a binomial experiment, the probability of
exactly X successes in n trials is
n!
P( X ) 
 p X  q n X
(n  X )! X !
© Copyright McGraw-Hill 2004
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Binomial Distribution Properties
The mean, variance, and standard deviation of
a variable that has the binomial distribution
can be found by using the following formulas.
mean
 np
variance
2  n  p q
standard deviation
  n  p q
© Copyright McGraw-Hill 2004
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Other Types of Distributions

The multinomial distribution is similar to the
binomial distribution but has the advantage
of allowing one to compute probabilities when
there are more than two outcomes.

The multinomial distribution is a general
distribution, and the binomial distribution is
a special case of the multinomial distribution.
© Copyright McGraw-Hill 2004
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Homework

Homework Chapter 5, Review Exercises
Page 276
 1,2,3,8,10,12,16,17
© Copyright McGraw-Hill 2004
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