Transcript of Geometric Distributions

Geometric
Distributions
Section 8.2
The 4 “commandments” of
Geometric Distributions
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Only 2 outcomes for each trial:
success or failure.
Same probability of success for
each trial
All independent trials.
The count of interest is the # of
trials required to get the first
success.
The geometric setting is
somewhat similar to the
binomial setting, the basic
difference being that the
geometric setting does not have
a fixed number of observations.
The probabilities of a geometric
sequence
Rule for calculating geometric probability
If X has a geometric distribution with probability p of success and (1-p) of
failure on each observation, the possible values of X are 1,2,3… If n is any
of these values, then the probability that the first success occurs on the nth
trial is…
P( X=n) = (1-p)n-1p
X
1
2
3
4
P(X)
p
(1-p)p
(1-p)2p (1-p)3p

1
p
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Geometric cont’d
If X is a geometric random variable with probability of
success p on each trial. The mean (expected value) of
the random variable, the expected number of trials
required to get the first success, is
1

p
1

p
The variance is  X2  2
p
1 p
So the standard deviation is  X 
p2
The probability that it takes more than n trials to see
the first success is:
P(X>n) = (1-p)n
Calculator
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geometpdf(p,X) p being the probability of
success and X being the number of the trial
which the first success occurs.
goemetcdf(p,X) where X is the maximum
number of trials to get the first success
Example
The pitcher is starting a new inning. The
probability that he’ll throw a strike is still .6 .
What is the expected number of pitches needed to
pitch a strike? 1/p≈1.6667 pitches
What is the probability that he will throw his first
strike on the 5th pitch? geometpdf(0.6, 5)=0.01536
What is the probability that it will take more than 5
pitches before he throws his first strike? 0.45=0.01024
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1 – geometcdf(0.6,5)=0.01024
Conclusion
Binomial setting is for a fixed number of trials whereas the
geometric setting has no fixed number of trials.
Review the 4 commandments! Learn them, accept them, embrace
them!
There are only 2 outcomes, success or failure and they each have a
fixed probability.
What is the shape of a geometpdf probability histogram when the
probability of success is 0.3? p=0.5? p=0.8? Right skewed
What is the shape of a geometcdf probability histogram? Left skewed