18.440: Lecture 12
Poisson random variables
Scott Sheffield
MIT
18.440 Lecture 12
Outline
Poisson random variable definition
Poisson random variable properties
Poisson random variable problems
18.440 Lecture 12
Outline
Poisson random variable definition
Poisson random variable properties
Poisson random variable problems
18.440 Lecture 12
Poisson random variables: motivating questions
I
How many raindrops hit a given square inch of sidewalk
during a ten minute period?
18.440 Lecture 12
Poisson random variables: motivating questions
I
How many raindrops hit a given square inch of sidewalk
during a ten minute period?
I
How many people fall down the stairs in a major city on a
given day?
18.440 Lecture 12
Poisson random variables: motivating questions
I
How many raindrops hit a given square inch of sidewalk
during a ten minute period?
I
How many people fall down the stairs in a major city on a
given day?
I
How many plane crashes in a given year?
18.440 Lecture 12
Poisson random variables: motivating questions
I
How many raindrops hit a given square inch of sidewalk
during a ten minute period?
I
How many people fall down the stairs in a major city on a
given day?
I
How many plane crashes in a given year?
I
How many radioactive particles emitted during a time period
in which the expected number emitted is 5?
18.440 Lecture 12
Poisson random variables: motivating questions
I
How many raindrops hit a given square inch of sidewalk
during a ten minute period?
I
How many people fall down the stairs in a major city on a
given day?
I
How many plane crashes in a given year?
I
How many radioactive particles emitted during a time period
in which the expected number emitted is 5?
I
How many calls to call center during a given minute?
18.440 Lecture 12
Poisson random variables: motivating questions
I
How many raindrops hit a given square inch of sidewalk
during a ten minute period?
I
How many people fall down the stairs in a major city on a
given day?
I
How many plane crashes in a given year?
I
How many radioactive particles emitted during a time period
in which the expected number emitted is 5?
I
How many calls to call center during a given minute?
I
How many goals scored during a 90 minute soccer game?
18.440 Lecture 12
Poisson random variables: motivating questions
I
How many raindrops hit a given square inch of sidewalk
during a ten minute period?
I
How many people fall down the stairs in a major city on a
given day?
I
How many plane crashes in a given year?
I
How many radioactive particles emitted during a time period
in which the expected number emitted is 5?
I
How many calls to call center during a given minute?
I
How many goals scored during a 90 minute soccer game?
I
How many notable gaffes during 90 minute debate?
18.440 Lecture 12
Poisson random variables: motivating questions
I
How many raindrops hit a given square inch of sidewalk
during a ten minute period?
I
How many people fall down the stairs in a major city on a
given day?
I
How many plane crashes in a given year?
I
How many radioactive particles emitted during a time period
in which the expected number emitted is 5?
I
How many calls to call center during a given minute?
I
How many goals scored during a 90 minute soccer game?
I
How many notable gaffes during 90 minute debate?
I
Key idea for all these examples: Divide time into large
number of small increments. Assume that during each
increment, there is some small probability of thing happening
(independently of other increments).
Remember what e is?
I
The number e is defined by e = limn→∞ (1 + 1/n)n .
18.440 Lecture 12
Remember what e is?
I
The number e is defined by e = limn→∞ (1 + 1/n)n .
I
It’s the amount of money that one dollar grows to over a year
when you have an interest rate of 100 percent, continuously
compounded.
18.440 Lecture 12
Remember what e is?
I
The number e is defined by e = limn→∞ (1 + 1/n)n .
I
It’s the amount of money that one dollar grows to over a year
when you have an interest rate of 100 percent, continuously
compounded.
I
Similarly, e λ = limn→∞ (1 + λ/n)n .
18.440 Lecture 12
Remember what e is?
I
The number e is defined by e = limn→∞ (1 + 1/n)n .
I
It’s the amount of money that one dollar grows to over a year
when you have an interest rate of 100 percent, continuously
compounded.
I
Similarly, e λ = limn→∞ (1 + λ/n)n .
I
It’s the amount of money that one dollar grows to over a year
when you have an interest rate of 100λ percent, continuously
compounded.
18.440 Lecture 12
Remember what e is?
I
The number e is defined by e = limn→∞ (1 + 1/n)n .
I
It’s the amount of money that one dollar grows to over a year
when you have an interest rate of 100 percent, continuously
compounded.
I
Similarly, e λ = limn→∞ (1 + λ/n)n .
I
It’s the amount of money that one dollar grows to over a year
when you have an interest rate of 100λ percent, continuously
compounded.
I
It’s also the amount of money that one dollar grows to over λ
years when you have an interest rate of 100 percent,
continuously compounded.
18.440 Lecture 12
Remember what e is?
I
The number e is defined by e = limn→∞ (1 + 1/n)n .
I
It’s the amount of money that one dollar grows to over a year
when you have an interest rate of 100 percent, continuously
compounded.
I
Similarly, e λ = limn→∞ (1 + λ/n)n .
I
It’s the amount of money that one dollar grows to over a year
when you have an interest rate of 100λ percent, continuously
compounded.
I
It’s also the amount of money that one dollar grows to over λ
years when you have an interest rate of 100 percent,
continuously compounded.
I
Can also change sign: e −λ = limn→∞ (1 − λ/n)n .
18.440 Lecture 12
Bernoulli random variable with n large and np = λ
I
Let λ be some moderate-sized number. Say λ = 2 or λ = 3.
Let n be a huge number, say n = 106 .
Bernoulli random variable with n large and np = λ
I
Let λ be some moderate-sized number. Say λ = 2 or λ = 3.
Let n be a huge number, say n = 106 .
I
Suppose I have a coin that comes up heads with probability
λ/n and I toss it n times.
Bernoulli random variable with n large and np = λ
I
Let λ be some moderate-sized number. Say λ = 2 or λ = 3.
Let n be a huge number, say n = 106 .
I
Suppose I have a coin that comes up heads with probability
λ/n and I toss it n times.
I
How many heads do I expect to see?
Bernoulli random variable with n large and np = λ
I
Let λ be some moderate-sized number. Say λ = 2 or λ = 3.
Let n be a huge number, say n = 106 .
I
Suppose I have a coin that comes up heads with probability
λ/n and I toss it n times.
I
How many heads do I expect to see?
I
Answer: np = λ.
Bernoulli random variable with n large and np = λ
I
Let λ be some moderate-sized number. Say λ = 2 or λ = 3.
Let n be a huge number, say n = 106 .
I
Suppose I have a coin that comes up heads with probability
λ/n and I toss it n times.
I
How many heads do I expect to see?
I
Answer: np = λ.
I
Let k be some moderate sized number (say k = 4). What is
the probability that I see exactly k heads?
Bernoulli random variable with n large and np = λ
I
Let λ be some moderate-sized number. Say λ = 2 or λ = 3.
Let n be a huge number, say n = 106 .
I
Suppose I have a coin that comes up heads with probability
λ/n and I toss it n times.
I
How many heads do I expect to see?
I
Answer: np = λ.
I
Let k be some moderate sized number (say k = 4). What is
the probability that I see exactly k heads?
I
Binomial
formula:
n k
p
(1
−
p)n−k =
k
18.440 Lecture 12
n(n−1)(n−2)...(n−k+1) k
p (1
k!
− p)n−k .
Bernoulli random variable with n large and np = λ
I
Let λ be some moderate-sized number. Say λ = 2 or λ = 3.
Let n be a huge number, say n = 106 .
I
Suppose I have a coin that comes up heads with probability
λ/n and I toss it n times.
I
How many heads do I expect to see?
I
Answer: np = λ.
I
Let k be some moderate sized number (say k = 4). What is
the probability that I see exactly k heads?
I
Binomial
formula:
n k
p
(1
−
p)n−k =
k
I
This is
n(n−1)(n−2)...(n−k+1) k
p (1 −
k!
k
k
approximately λk! (1 − p)n−k ≈ λk! e −λ .
p)n−k .
Bernoulli random variable with n large and np = λ
I
Let λ be some moderate-sized number. Say λ = 2 or λ = 3.
Let n be a huge number, say n = 106 .
I
Suppose I have a coin that comes up heads with probability
λ/n and I toss it n times.
I
How many heads do I expect to see?
I
Answer: np = λ.
I
Let k be some moderate sized number (say k = 4). What is
the probability that I see exactly k heads?
I
Binomial
formula:
n k
p
(1
−
p)n−k =
k
n(n−1)(n−2)...(n−k+1) k
p (1 −
k!
k
k
approximately λk! (1 − p)n−k ≈ λk! e −λ .
p)n−k .
I
This is
I
A Poisson random variable X with parameter λ satisfies
k
P{X = k} = λk! e −λ for integer k ≥ 0.
Outline
Poisson random variable definition
Poisson random variable properties
Poisson random variable problems
18.440 Lecture 12
Outline
Poisson random variable definition
Poisson random variable properties
Poisson random variable problems
18.440 Lecture 12
Probabilities sum to one
I
A Poisson random variable X with parameter λ satisfies
k
p(k) = P{X = k} = λk! e −λ for integer k ≥ 0.
Probabilities sum to one
I
I
A Poisson random variable X with parameter λ satisfies
k
p(k) = P{X = k} = λk! e −λ for integer k ≥ 0.
P
How can we show that ∞
k=0 p(k) = 1?
Probabilities sum to one
I
I
I
A Poisson random variable X with parameter λ satisfies
k
p(k) = P{X = k} = λk! e −λ for integer k ≥ 0.
P
How can we show that ∞
k=0 p(k) = 1?
P
λk
Use Taylor expansion e λ = ∞
k=0 k! .
18.440 Lecture 12
Expectation
I
A Poisson random variable X with parameter λ satisfies
k
P{X = k} = λk! e −λ for integer k ≥ 0.
Expectation
I
A Poisson random variable X with parameter λ satisfies
k
P{X = k} = λk! e −λ for integer k ≥ 0.
I
What is E [X ]?
Expectation
I
A Poisson random variable X with parameter λ satisfies
k
P{X = k} = λk! e −λ for integer k ≥ 0.
I
What is E [X ]?
I
We think of a Poisson random variable as being (roughly) a
Bernoulli (n, p) random variable with n very large and
p = λ/n.
Expectation
I
A Poisson random variable X with parameter λ satisfies
k
P{X = k} = λk! e −λ for integer k ≥ 0.
I
What is E [X ]?
I
We think of a Poisson random variable as being (roughly) a
Bernoulli (n, p) random variable with n very large and
p = λ/n.
I
This would suggest E [X ] = λ. Can we show this directly from
the formula for P{X = k}?
Expectation
I
A Poisson random variable X with parameter λ satisfies
k
P{X = k} = λk! e −λ for integer k ≥ 0.
I
What is E [X ]?
I
We think of a Poisson random variable as being (roughly) a
Bernoulli (n, p) random variable with n very large and
p = λ/n.
I
This would suggest E [X ] = λ. Can we show this directly from
the formula for P{X = k}?
I
By definition of expectation
E [X ] =
∞
X
k=0
P{X = k}k =
∞
∞
X
X
λk
λk
k e −λ =
e −λ .
k!
(k − 1)!
k=0
k=1
Expectation
I
A Poisson random variable X with parameter λ satisfies
k
P{X = k} = λk! e −λ for integer k ≥ 0.
I
What is E [X ]?
I
We think of a Poisson random variable as being (roughly) a
Bernoulli (n, p) random variable with n very large and
p = λ/n.
I
This would suggest E [X ] = λ. Can we show this directly from
the formula for P{X = k}?
I
By definition of expectation
E [X ] =
∞
X
P{X = k}k =
k=0
I
Setting j = k − 1, this is λ
∞
∞
X
X
λk
λk
k e −λ =
e −λ .
k!
(k − 1)!
k=0
λj −λ
j=0 j! e
P∞
k=1
= λ.
Variance
I
Given P{X = k} =
18.440 Lecture 12
λk −λ
k! e
for integer k ≥ 0, what is Var[X ]?
Variance
I
I
k
Given P{X = k} = λk! e −λ for integer k ≥ 0, what is Var[X ]?
Think of X as (roughly) a Bernoulli (n, p) random variable
with n very large and p = λ/n.
18.440 Lecture 12
Variance
I
I
I
k
Given P{X = k} = λk! e −λ for integer k ≥ 0, what is Var[X ]?
Think of X as (roughly) a Bernoulli (n, p) random variable
with n very large and p = λ/n.
This suggests Var[X ] ≈ npq ≈ λ (since np ≈ λ and
q = 1 − p ≈ 1). Can we show directly that Var[X ] = λ?
18.440 Lecture 12
Variance
I
I
I
k
Given P{X = k} = λk! e −λ for integer k ≥ 0, what is Var[X ]?
Think of X as (roughly) a Bernoulli (n, p) random variable
with n very large and p = λ/n.
This suggests Var[X ] ≈ npq ≈ λ (since np ≈ λ and
q = 1 − p ≈ 1). Can we show directly that Var[X ] = λ?
18.440 Lecture 12
Variance
I
I
I
I
k
Given P{X = k} = λk! e −λ for integer k ≥ 0, what is Var[X ]?
Think of X as (roughly) a Bernoulli (n, p) random variable
with n very large and p = λ/n.
This suggests Var[X ] ≈ npq ≈ λ (since np ≈ λ and
q = 1 − p ≈ 1). Can we show directly that Var[X ] = λ?
Compute
2
E [X ] =
∞
X
k=0
18.440 Lecture 12
2
P{X = k}k =
∞
X
k=0
k
k
2λ
−λ
k!
e
=λ
∞
X
k=1
k
λk−1 −λ
e .
(k − 1)!
Variance
I
I
I
I
k
Given P{X = k} = λk! e −λ for integer k ≥ 0, what is Var[X ]?
Think of X as (roughly) a Bernoulli (n, p) random variable
with n very large and p = λ/n.
This suggests Var[X ] ≈ npq ≈ λ (since np ≈ λ and
q = 1 − p ≈ 1). Can we show directly that Var[X ] = λ?
Compute
2
E [X ] =
∞
X
P{X = k}k =
k=0
I
2
∞
X
k=0
k
k
2λ
−λ
k!
e
=λ
∞
X
k=1
k
λk−1 −λ
e .
(k − 1)!
Setting j = k − 1, this is


∞
j
X
λ
λ  (j + 1) e −λ  = λE [X + 1] = λ(λ + 1).
j!
j=0
18.440 Lecture 12
Variance
I
I
I
I
k
Given P{X = k} = λk! e −λ for integer k ≥ 0, what is Var[X ]?
Think of X as (roughly) a Bernoulli (n, p) random variable
with n very large and p = λ/n.
This suggests Var[X ] ≈ npq ≈ λ (since np ≈ λ and
q = 1 − p ≈ 1). Can we show directly that Var[X ] = λ?
Compute
2
E [X ] =
∞
X
P{X = k}k =
k=0
I
2
∞
X
k=0
k
k
2λ
−λ
k!
e
=λ
∞
X
k=1
k
λk−1 −λ
e .
(k − 1)!
Setting j = k − 1, this is


∞
j
X
λ
λ  (j + 1) e −λ  = λE [X + 1] = λ(λ + 1).
j!
j=0
I
Then Var[X ] = E [X 2 ] − E [X ]2 = λ(λ + 1) − λ2 = λ.
Outline
Poisson random variable definition
Poisson random variable properties
Poisson random variable problems
18.440 Lecture 12
Outline
Poisson random variable definition
Poisson random variable properties
Poisson random variable problems
18.440 Lecture 12
Poisson random variable problems
I
A country has an average of 2 plane crashes per year.
18.440 Lecture 12
Poisson random variable problems
I
A country has an average of 2 plane crashes per year.
I
How reasonable is it to assume the number of crashes is
Poisson with parameter 2?
18.440 Lecture 12
Poisson random variable problems
I
A country has an average of 2 plane crashes per year.
I
How reasonable is it to assume the number of crashes is
Poisson with parameter 2?
I
Assuming this, what is the probability of exactly 2 crashes?
Of zero crashes? Of four crashes?
18.440 Lecture 12
Poisson random variable problems
I
A country has an average of 2 plane crashes per year.
I
How reasonable is it to assume the number of crashes is
Poisson with parameter 2?
I
Assuming this, what is the probability of exactly 2 crashes?
Of zero crashes? Of four crashes?
I
A city has an average of five major earthquakes a century.
What is the probability that there is an earthquake in a given
decade (assuming the number of earthquakes per decade is
Poisson)?
18.440 Lecture 12
Poisson random variable problems
I
A country has an average of 2 plane crashes per year.
I
How reasonable is it to assume the number of crashes is
Poisson with parameter 2?
I
Assuming this, what is the probability of exactly 2 crashes?
Of zero crashes? Of four crashes?
I
A city has an average of five major earthquakes a century.
What is the probability that there is an earthquake in a given
decade (assuming the number of earthquakes per decade is
Poisson)?
I
If both candidates average one major gaffe per debate, what is
the probably that the first has at least one major gaffe and
the second doesn’t? (What assumptions are we making?)
18.440 Lecture 12
Poisson random variable problems
I
A country has an average of 2 plane crashes per year.
I
How reasonable is it to assume the number of crashes is
Poisson with parameter 2?
I
Assuming this, what is the probability of exactly 2 crashes?
Of zero crashes? Of four crashes?
I
A city has an average of five major earthquakes a century.
What is the probability that there is an earthquake in a given
decade (assuming the number of earthquakes per decade is
Poisson)?
I
If both candidates average one major gaffe per debate, what is
the probably that the first has at least one major gaffe and
the second doesn’t? (What assumptions are we making?)
I
A casino deals one million five-card poker hands per year.
Approximate the probability that there are exactly 2 royal
flush hands during a given year.
18.440 Lecture 12