Geometric
Distribution
Lecture 14
Geometric Distribution
Text: A Course in Probability by Weiss 5.6
STAT 225 Introduction to Probability Models
February 23, 2014
Whitney Huang
Purdue University
14.1
Agenda
Geometric
Distribution
14.2
Review
Geometric
Distribution
So far, we have covered
Bernoulli and Binomial distribution: number of successes
in an independent trials (sampling with replacement) with
fixed sample size
14.3
Review
Geometric
Distribution
So far, we have covered
Bernoulli and Binomial distribution: number of successes
in an independent trials (sampling with replacement) with
fixed sample size
Hypergeometric distribution: number of successes in a
dependent trials (sampling without replacement) with fixed
sample size
14.3
Review
Geometric
Distribution
So far, we have covered
Bernoulli and Binomial distribution: number of successes
in an independent trials (sampling with replacement) with
fixed sample size
Hypergeometric distribution: number of successes in a
dependent trials (sampling without replacement) with fixed
sample size
Poisson distribution: number of successes (events)
occurring in a fixed interval of time and/or space without
fixed sample size
14.3
Review
Geometric
Distribution
So far, we have covered
Bernoulli and Binomial distribution: number of successes
in an independent trials (sampling with replacement) with
fixed sample size
Hypergeometric distribution: number of successes in a
dependent trials (sampling without replacement) with fixed
sample size
Poisson distribution: number of successes (events)
occurring in a fixed interval of time and/or space without
fixed sample size
14.3
Review
Geometric
Distribution
So far, we have covered
Bernoulli and Binomial distribution: number of successes
in an independent trials (sampling with replacement) with
fixed sample size
Hypergeometric distribution: number of successes in a
dependent trials (sampling without replacement) with fixed
sample size
Poisson distribution: number of successes (events)
occurring in a fixed interval of time and/or space without
fixed sample size
In some cases, we want to know the sample size necessary to
get a certain number of successes
Geometric distribution: number of trials until the 1st
success (including the success)
14.3
Review
Geometric
Distribution
So far, we have covered
Bernoulli and Binomial distribution: number of successes
in an independent trials (sampling with replacement) with
fixed sample size
Hypergeometric distribution: number of successes in a
dependent trials (sampling without replacement) with fixed
sample size
Poisson distribution: number of successes (events)
occurring in a fixed interval of time and/or space without
fixed sample size
In some cases, we want to know the sample size necessary to
get a certain number of successes
Geometric distribution: number of trials until the 1st
success (including the success)
Negative Binomial distribution: number of trials until the rth
success (including the rth success)
14.3
Review
Geometric
Distribution
So far, we have covered
Bernoulli and Binomial distribution: number of successes
in an independent trials (sampling with replacement) with
fixed sample size
Hypergeometric distribution: number of successes in a
dependent trials (sampling without replacement) with fixed
sample size
Poisson distribution: number of successes (events)
occurring in a fixed interval of time and/or space without
fixed sample size
In some cases, we want to know the sample size necessary to
get a certain number of successes
Geometric distribution: number of trials until the 1st
success (including the success)
Negative Binomial distribution: number of trials until the rth
success (including the rth success)
In both Geometric and Binomial distribution, the trials are
independent
14.3
Geometric Distribution:
Geometric
Distribution
Characteristics of the Geometric Distribution:
Let X be a Geometric r.v.
The definition of X : The number of trials it takes to get the
1st success
14.4
Geometric Distribution:
Geometric
Distribution
Characteristics of the Geometric Distribution:
Let X be a Geometric r.v.
The definition of X : The number of trials it takes to get the
1st success
The support: x = 1, 2, · · ·
14.4
Geometric Distribution:
Geometric
Distribution
Characteristics of the Geometric Distribution:
Let X be a Geometric r.v.
The definition of X : The number of trials it takes to get the
1st success
The support: x = 1, 2, · · ·
Its parameter(s) and definition(s): p: the probability of
success in a single trial
14.4
Geometric Distribution:
Geometric
Distribution
Characteristics of the Geometric Distribution:
Let X be a Geometric r.v.
The definition of X : The number of trials it takes to get the
1st success
The support: x = 1, 2, · · ·
Its parameter(s) and definition(s): p: the probability of
success in a single trial
The probability mass function (pmf):
pX (x) = p(1 − p)x−1 for x = 1, 2, · · ·
14.4
Geometric
Distribution
Geometric Distribution:
Characteristics of the Geometric Distribution:
Let X be a Geometric r.v.
The definition of X : The number of trials it takes to get the
1st success
The support: x = 1, 2, · · ·
Its parameter(s) and definition(s): p: the probability of
success in a single trial
The probability mass function (pmf):
pX (x) = p(1 − p)x−1 for x = 1, 2, · · ·
The expected value: E[X ] =
1
p
14.4
Geometric
Distribution
Geometric Distribution:
Characteristics of the Geometric Distribution:
Let X be a Geometric r.v.
The definition of X : The number of trials it takes to get the
1st success
The support: x = 1, 2, · · ·
Its parameter(s) and definition(s): p: the probability of
success in a single trial
The probability mass function (pmf):
pX (x) = p(1 − p)x−1 for x = 1, 2, · · ·
The expected value: E[X ] =
The variance: Var (X ) =
1
p
1−p
p2
14.4
Properties of Geometric distribution
Geometric
Distribution
Tail Probability: P(X > x) = (1 − p)x
Example
14.5
Properties of Geometric distribution
Geometric
Distribution
Tail Probability: P(X > x) = (1 − p)x
Example
P(X ≥ 4) = P(X > 3) = (1 − p)3
14.5
Properties of Geometric distribution
Geometric
Distribution
Tail Probability: P(X > x) = (1 − p)x
Example
P(X ≥ 4) = P(X > 3) = (1 − p)3
P(3 ≤ X ≤ 9) = P(X ≥ 3) − P(X ≥ 10) = P(X >
2) − P(X > 9) = (1 − p)2 − (1 − p)9
14.5
Properties of Geometric distribution
Geometric
Distribution
Tail Probability: P(X > x) = (1 − p)x
Example
P(X ≥ 4) = P(X > 3) = (1 − p)3
P(3 ≤ X ≤ 9) = P(X ≥ 3) − P(X ≥ 10) = P(X >
2) − P(X > 9) = (1 − p)2 − (1 − p)9
Memoryless Property: P(X > s + t|X > s) = P(X > t)
Example
14.5
Properties of Geometric distribution
Geometric
Distribution
Tail Probability: P(X > x) = (1 − p)x
Example
P(X ≥ 4) = P(X > 3) = (1 − p)3
P(3 ≤ X ≤ 9) = P(X ≥ 3) − P(X ≥ 10) = P(X >
2) − P(X > 9) = (1 − p)2 − (1 − p)9
Memoryless Property: P(X > s + t|X > s) = P(X > t)
Example
P(X > 15|X > 9) = P(X > 6) = (1 − p)6
14.5
Properties of Geometric distribution
Geometric
Distribution
Tail Probability: P(X > x) = (1 − p)x
Example
P(X ≥ 4) = P(X > 3) = (1 − p)3
P(3 ≤ X ≤ 9) = P(X ≥ 3) − P(X ≥ 10) = P(X >
2) − P(X > 9) = (1 − p)2 − (1 − p)9
Memoryless Property: P(X > s + t|X > s) = P(X > t)
Example
P(X > 15|X > 9) = P(X > 6) = (1 − p)6
P(X ≤ 8|X ≥ 5) = 1 − P(X > 8|X ≥ 5) = 1 − P(X >
8|X > 6) = 1 − P(X > 2) = 1 − (1 − p)2
14.5
Example 36
Geometric
Distribution
Suppose Dunphy is really bad at tossing a Frisbee. Suppose
Dunphy hits pedestrians at a rate of 1 out of 5 people that walk
past the campus mall. Let X be of number of tosses it takes to
hit the 1st pedestrian.
1
Name distribution (with parameter(s))
14.6
Example 36
Geometric
Distribution
Suppose Dunphy is really bad at tossing a Frisbee. Suppose
Dunphy hits pedestrians at a rate of 1 out of 5 people that walk
past the campus mall. Let X be of number of tosses it takes to
hit the 1st pedestrian.
1
2
Name distribution (with parameter(s))
What is the probability that his first accidental hitting is the
5th person that walks by?
14.6
Example 36
Geometric
Distribution
Suppose Dunphy is really bad at tossing a Frisbee. Suppose
Dunphy hits pedestrians at a rate of 1 out of 5 people that walk
past the campus mall. Let X be of number of tosses it takes to
hit the 1st pedestrian.
1
2
3
Name distribution (with parameter(s))
What is the probability that his first accidental hitting is the
5th person that walks by?
What is the probability that he does not hit anyone in the
first 10 tosses?
14.6
Example 36
Geometric
Distribution
Suppose Dunphy is really bad at tossing a Frisbee. Suppose
Dunphy hits pedestrians at a rate of 1 out of 5 people that walk
past the campus mall. Let X be of number of tosses it takes to
hit the 1st pedestrian.
1
2
Name distribution (with parameter(s))
What is the probability that his first accidental hitting is the
5th person that walks by?
3
What is the probability that he does not hit anyone in the
first 10 tosses?
4
What is the probability that it takes no more than 20 tosses
to hit the first pedestrian given that he does not hit anyone
in the first 10 tosses?
14.6
Example 36 cont’d
Geometric
Distribution
Solution.
1
X ∼ Geo(p = .2)
14.7
Example 36 cont’d
Geometric
Distribution
Solution.
1
X ∼ Geo(p = .2)
2
P(X = 5) = (0.2)(1 − 0.2)4 = 0.0819
14.7
Example 36 cont’d
Geometric
Distribution
Solution.
1
X ∼ Geo(p = .2)
2
P(X = 5) = (0.2)(1 − 0.2)4 = 0.0819
3
P(X > 10) = (1 − .2)10 = 0.1074
14.7
Geometric
Distribution
Example 36 cont’d
Solution.
1
X ∼ Geo(p = .2)
2
P(X = 5) = (0.2)(1 − 0.2)4 = 0.0819
3
4
P(X > 10) = (1 − .2)10 = 0.1074
P(X ≤ 20|X > 10) = 1 − P(X > 20|X >
10)
memoryless propetry
=====
1 − P(X > 10) = 1 − .1073 = 0.8926
14.7
Practice HW 2, Problem 12
Geometric
Distribution
You randomly call friends who could be potential partners for a
dance. You think that they all respond to your requests
independently of each other, and you estimate that each one is
7% likely to accept your request. Let X denote the number of
calls to successfully get a date.
1
What are the distribution and parameters of X ?
14.8
Practice HW 2, Problem 12
Geometric
Distribution
You randomly call friends who could be potential partners for a
dance. You think that they all respond to your requests
independently of each other, and you estimate that each one is
7% likely to accept your request. Let X denote the number of
calls to successfully get a date.
1
What are the distribution and parameters of X ?
2
How many calls do you expect to make to get a date?
14.8
Practice HW 2, Problem 12
Geometric
Distribution
You randomly call friends who could be potential partners for a
dance. You think that they all respond to your requests
independently of each other, and you estimate that each one is
7% likely to accept your request. Let X denote the number of
calls to successfully get a date.
1
What are the distribution and parameters of X ?
2
How many calls do you expect to make to get a date?
3
What is the probability you get a date on the 12th call?
14.8
Practice HW 2, Problem 12
Geometric
Distribution
You randomly call friends who could be potential partners for a
dance. You think that they all respond to your requests
independently of each other, and you estimate that each one is
7% likely to accept your request. Let X denote the number of
calls to successfully get a date.
1
What are the distribution and parameters of X ?
2
How many calls do you expect to make to get a date?
3
What is the probability you get a date on the 12th call?
4
What is the probability it takes you at least 10 calls to get a
date?
14.8
Practice HW 2, Problem 12
Geometric
Distribution
You randomly call friends who could be potential partners for a
dance. You think that they all respond to your requests
independently of each other, and you estimate that each one is
7% likely to accept your request. Let X denote the number of
calls to successfully get a date.
1
What are the distribution and parameters of X ?
2
How many calls do you expect to make to get a date?
3
What is the probability you get a date on the 12th call?
4
What is the probability it takes you at least 10 calls to get a
date?
5
What is the probability it takes you no more than 15 calls
to get a date?
14.8
Practice HW 2, Problem 12
Geometric
Distribution
You randomly call friends who could be potential partners for a
dance. You think that they all respond to your requests
independently of each other, and you estimate that each one is
7% likely to accept your request. Let X denote the number of
calls to successfully get a date.
1
What are the distribution and parameters of X ?
2
How many calls do you expect to make to get a date?
3
What is the probability you get a date on the 12th call?
4
What is the probability it takes you at least 10 calls to get a
date?
5
What is the probability it takes you no more than 15 calls
to get a date?
Given you’ve already called 8 people and still do not have
a date, what is the probability it will take less than 14 calls?
6
14.8
Practice HW 2, Problem 12
Geometric
Distribution
Solution.
1
X ∼ Geo(p = 0.07)
14.9
Practice HW 2, Problem 12
Geometric
Distribution
Solution.
1
X ∼ Geo(p = 0.07)
2
E[X ] =
1
p
=
1
0.07
= 14.2857
14.9
Practice HW 2, Problem 12
Geometric
Distribution
Solution.
1
X ∼ Geo(p = 0.07)
2
E[X ] =
3
P(X = 12) = (0.07)(1 − 0.07)11 = 0.0315
1
p
=
1
0.07
= 14.2857
14.9
Practice HW 2, Problem 12
Geometric
Distribution
Solution.
1
X ∼ Geo(p = 0.07)
2
E[X ] =
3
P(X = 12) = (0.07)(1 − 0.07)11 = 0.0315
4
P(X ≥ 10) = P(X > 9) = (1 − 0.07)9 = 0.5204
1
p
=
1
0.07
= 14.2857
14.9
Practice HW 2, Problem 12
Geometric
Distribution
Solution.
1
X ∼ Geo(p = 0.07)
2
E[X ] =
3
P(X = 12) = (0.07)(1 − 0.07)11 = 0.0315
4
P(X ≥ 10) = P(X > 9) = (1 − 0.07)9 = 0.5204
5
P(X ≤ 15) = 1 − P(X > 15) = 1 − (1 − 0.07)15 = 0.6633
1
p
=
1
0.07
= 14.2857
14.9
Practice HW 2, Problem 12
Geometric
Distribution
Solution.
1
X ∼ Geo(p = 0.07)
2
E[X ] =
3
P(X = 12) = (0.07)(1 − 0.07)11 = 0.0315
4
P(X ≥ 10) = P(X > 9) = (1 − 0.07)9 = 0.5204
5
P(X ≤ 15) = 1 − P(X > 15) = 1 − (1 − 0.07)15 = 0.6633
6
1
p
=
1
0.07
= 14.2857
P(X < 14|X > 8) = 1 − P(X > 13|X > 8) = 1 − P(X >
5) = 1 − (1 − 0.07)5 = 0.3043
14.9
Geometric
Distribution
Summary
In today’s lecture, we
Introduced the Geometric distribution
Tail Probability
Memoryless Property
14.10