Transcript Probability Distributions

Probability Distributions
Chapter 6
McGraw-Hill/Irwin
©The McGraw-Hill Companies, Inc. 2008
GOALS
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Define the terms probability distribution and random
variable.
Distinguish between discrete and continuous
probability distributions.
Calculate the mean, variance, and standard deviation
of a discrete probability distribution.
Describe the characteristics of and compute probabilities
using the binomial probability distribution.
Describe the characteristics of and compute probabilities
using the hypergeometric probability distribution.- skip
Describe the characteristics of and compute probabilities
using the Poisson distribution.
Random Variables
Random variable a quantity resulting from an
experiment that, by chance, can assume different
values.
(eg) The number of head in three coin tosses.
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What is a Probability Distribution?
Experiment: Toss a
coin three times.
Observe the number
of heads. The
possible results are:
zero, one, two, and
three heads.
What is the probability
distribution for the
number of heads?
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Probability Distribution of Number of
Heads Observed in 3 Tosses of a Coin
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Characteristics of a Probability
Distribution
A random variable can be described by its probability
distribution.
Probability distribution is similar to the relative
frequency distribution * difference: past vs future
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Types of Random Variables
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Discrete Random Variable can assume only
certain clearly separated values. It is usually
the result of counting something (not always).
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Continuous Random Variable can assume
an infinite number of values within a given
range. It is usually the result of some type of
measurement
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Some may have decimal values (7.2, 8.9, --).
Width, height, air pressure, etc.
Probability distribution (discrete r.v.) &
probability density function (continuous r.v.)
Discrete Random Variables - Examples
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The number of students in a class.
The number of children in a family.
The number of cars passing through a toll gate in an
hour.
Continuous Random Variables Examples
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The distance students travel to class.
The time it takes an executive to drive to
work.
The length of an afternoon nap.
The length of time of a particular phone call.
Features of a Discrete Probability
Distribution
The main features of a discrete probability
distribution are:
 The sum of the probabilities of the all
outcomes is 1.00.
 The probability of a particular outcome is
between 0 and 1.00.
 The outcomes are mutually exclusive.
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The Mean of a Probability Distribution
MEAN
•The mean is a typical value used to represent the
central location of a probability distribution.
•The mean of a probability distribution is also
referred to as its expected value.
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The Variance, and Standard
Deviation of a Probability Distribution
Variance and Standard Deviation
• Measures the degree of spread around its mean in a distribution
• The computational steps are:
1. Subtract the mean from each value, and square this difference.
2. Multiply each squared difference by its probability.
3. Sum the resulting products to arrive at the variance.
The standard deviation is the positive square root of the variance.
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Mean, Variance, and Standard Deviation of a
Probability Distribution (Random Variable)- Example
John, a car salesman, usually sells the
largest number of cars on Saturday. He
has developed the following probability
distribution for the number of cars he
expects to sell on a particular Saturday.
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Mean of a Probability Distribution (Random
Variable) - Example
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Variance and Standard Deviation of a Probability
Distribution (Random Variable) - Example
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Binomial Probability Distribution
Characteristics of a Binomial Probability
Distribution
 Outcome on each trial of an experiment is classified
into one of two mutually exclusive categories.
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The random variable counts the number of
‘successes’ in a fixed (total) number of trials.
The probability of success is the same for each trial.
Each trial is independent of any other trial.
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‘a success’ or ‘a failure’
The outcome of one trial does not affect the outcome of any
other trial.
Binomial Probability Formula
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Binomial Probability - Example
There are five flights
daily from Pittsburgh
to Boston. Suppose
the probability that
any flight arrives
late is .20.
What is the probability
that none (one) of
the flights is late
today?
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Binomial Probability - Excel
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Binomial Dist. – Mean and Variance
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Binomial Dist. – Mean and Variance:
Example
For the example
regarding the number
of late flights, recall
that  =.20 and n = 5.
What is the average
number of late flights?
What is the variance of
the number of late
flights?
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Binomial Dist. – Mean and Variance:
Another Solution
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Binomial Distribution- Table (pp. 794-798)
Five percent of the gears produced by a milling machine are
defective. What is the probability that out of six gears selected
at random none will be defective? Exactly one? Exactly two?
Exactly three? Exactly four? Exactly five? Exactly six out of six?
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Binomial – Shapes for Varying 
(n constant) * how the shape changes?
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Binomial – Shapes for Varying n
( constant) * how the shape changes?
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Cumulative Binomial Probability
Distributions
A study by the Department of Transportation concluded
that 76.2 percent of front seat occupants used seat
belts. A sample of 12 vehicles is selected. What is the
probability the front seat occupants in at least 7 of the
12 vehicles are wearing seat belts?
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Cumulative Binomial Probability
Distributions - Excel
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Poisson Probability Distribution
The Poisson probability distribution describes the
number of times some event occurs during a
specified interval.
- The interval may be time, distance, or area.
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Assumptions of the Poisson Distribution
(1)
(2)
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The probability is proportional to the length of the interval.
The intervals do not overlap and are independent.
Poisson distribution is a limiting form of the
binomial distribution, when  is very small & n is
very large.
Poisson Probability Distribution
The Poisson distribution can be
described mathematically using the
formula:
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Poisson Probability Distribution
mean number of successes 
can be determined in binomial
situations by n .
 The
where n is the number of trials and 
the probability of a success.
– The variance of the Poisson
distribution is also equal to n .
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Poisson Probability Distribution Example
Assume baggage is rarely lost by Korea Airlines.
Suppose a random sample of 1,000 flights shows a
total of 300 bags were lost. Thus, the arithmetic
mean of lost bags per flight is 0.3. If the number of
lost bags per flight follows a Poisson distribution with
u = 0.3, find the probability of not losing any bags in
a flight.
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Poisson Probability Distribution - Table
Suppose the number of lost bags per flight follows a Poisson distribution
with mean = 0.3. Find the probability of losing no bag, one bag, two
bags, --- , on a particular flight. (Table is available).
* When should the supervisor be suspicious about lost bags?
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Poisson distribution as an estimate of
binomial distribution (large n, small )
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Example: probability of a strong typoon striking a
certain area in any one year- .05 and homeowner
purchased a house there on a 30 year mortgage.
(Insurance company) Want to know the likelihood of
at least one typoon strike during the 30 year period.
Two approaches to calculate P(x ≥ 1) = 1 - P(x=0).
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Poisson distribution: (using u = n  = 1.5)
Binomial distribution: (using  = .05, n = 30)
End of Chapter 6
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