Transcript geometric distribution

Engineering Statistics ECIV 2305
Chapter 3
DISCRETE PROBABILITY DISTRIBUTIONS
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3.1 The Binomial Distribution
3.2 The Geometric Distribution
3.3 The Hypergeometric Distribution
3.4 The Poisson Distribution
3.2 The Geometric Distribution
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Engineering Statistics ECIV 2305
Section 3.2
The Geometric Distribution
3.2 The Geometric Distribution
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Recall that the Binomial distribution is the distribution
of the number of successes occurring in a fixed
number of trials n.
 Suppose that our interest is to count the number of
Bernoulli trials performed until the first success
occurs. Such random variable is said to have a
geometric distribution with parameter p.
 The pmf of the geometric distribution is:
P(X = x) = (1 – p)x-1p
E(X) = 1/p ,
Var(X) = (1 – p)/p2
 The cdf of the geometric distribution is:
P(X≤x) = 1 – (1 – p)x
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3.2 The Geometric Distribution
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Examples
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What is the probability that a Head is obtained for the first
time on the fourth coin toss?
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P(X = 4) = (1 – p)4-1p = (1 – 0.5)3(0.5) = 1/16
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What is the probability that a Head is obtained for the first
time on the first coin toss?
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What is the probability that a Head is obtained for the first
time on the first coin toss?
3.2 The Geometric Distribution
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Example
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Draw the pmf of a geometric distribution with parameter
p=1/2 and the pmf of a geometric distribution with
parameter p=1/5 . (Fig. 3.12 and 3.13 in your textbook)
3.2 The Geometric Distribution
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Example Air Force Planes
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(page 175)
What is the distribution of the number of attempts needed to
start a plane’s engine?
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This is a case of a geometric distribution since attention is directed
at the number of trials until the first success. A “success” here is
the event that the plane’s engines start ►► p = 0.75
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P(X = x)
3.2 The Geometric Distribution
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Example Telephone Ticket Sales
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(page 176)
People make telephone calls to a salesperson to buy tickets for
an event. The probability that a salesperson becomes free from
the previous customer is 0.1. In other words, P(reaching a
salesperson) = 0.1
Find the distribution and the expectation of the number of calls
that a person needs to make until a salesperson is reached?
What is the probability that a caller gets through on the fifth
attempt?
What is the probability that 15 or more calls are needed?
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The placing of a call is a Bernoulli trial with a success probability of
p=0.1
The geometric distribution is appropriate since the quantity of interest
is the number of calls made until the first success.
3.2 The Geometric Distribution
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… continued Telephone Ticket Sales …
3.2 The Geometric Distribution
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