Download Stat 260 - Lecture 12 Recap of Last Class • We introduced the

Stat 260 - Lecture 12
Recap of Last Class
• We introduced the binomial distribution and
binomial random variables.
Today:
• Begin with an example illustrating the use
of the binomial distribution.
• We will introduce another important distribution known as the Poisson distribution.
• Also introduce the Poisson process.
• Go through some examples...
The Poisson Distribution
• A discrete random variable X is said to
have a Poisson distribution with parameter λ > 0 if the pmf of X is
e−λλx
P (X = x) = p(x; λ) =
, x = 0, 1, 2, 3, ...
x!
• A Poisson random variable takes values on
the nonnegative integers.
• The distribution is used often in statistics
for modelling count data.
Simple Example:
If X ∼ Poisson(λ = 2) then what is
P (X > 0)?
P (X > 0) = 1 − P (X ≤ 0) = 1 − P (X = 0)
e−220
= 1 − p(0; 2) = 1 −
≈ 0.865
0!
Relationship to the binomial
• The Poisson distribution is often used as an
approximation to the binomial distribution.
• Proposition: Suppose X ∼ Bin(n, p) so
that the pmf is given by b(x; n; p). If we let
n → ∞ and p → 0 in such a way that np approaches a value λ > 0. Then b(x; n, p) →
p(x; λ).
• In other words, if X ∼ Bin(n, p) with n large
and p small, we can approximate the pmf
of X with the Poisson pmf p(x; λ = np).
• Rule of thumb to use this approximation:
n ≥ 100, p ≤ 0.01 and np ≤ 20.
Expected Value and Variance - Poisson
• If X has a Poisson distribution with parameter λ, then E[x] = λ and V [X] = λ.
• Note that for a Poisson random variable
E[x] = V [X].
The Poisson Process
• Counting Process: There are many situations in which we observe the occurrence
of events of a particular type over time.
For example:
1. In a soccer game, the events of interest might be goals scored, which occur
throughout the course of the game.
2. In a hospital emergency room, the events
of interest might be new hospital admissions that occur throughout the course
of the day.
3. At an airport, the events of interest might
be the arrival of airplanes during some
period of time.
In all these cases, we observe some process
which yields recurring events: goals, hospital admissions or airplane arrivals over time
and we observe the time at which each
event occurs.
• Often we are not interested in the exact
times at which events occur; instead, we
are interested in the total number of events
during some time interval of length t.
1. Soccer example: # goals in the 1st minute
of the 2nd half.
2. Hospital example: # of new admissions
between 5:30 PM and 6:00PM.
3. Airport example: # of airplanes arriving
during any 20 minute interval during the
day.
• In this case, we are interested in a random
process, N (t), which counts the number of
events in any interval of time having length
t.
• We call N (t) a counting process. For any
fixed t0, the quantity N (t0) is a random
variable.
• For example starting at t = 0, the counting
process N (t) might count the total number of points scored by Kobe Bryant after t minutes of a lakers game. So here,
N (12), N (24), N (36) and N (48) represent
the total number of points scored after
each quarter - each of these is a random
variable.
• Poisson Process: A counting process, N (t),
is called a Poisson process with rate parameter α if all of the following three conditions
hold:
1. The number of events in nonoverlapping
intervals occur independently of one another.
2. The probability distribution of the number of events in any interval of time depends only on the length of the interval.
3. The probability of x events in an interval
of length t is
P (N (t) = x) =
(αt)
−αt
e
x
x!
that is, for each t we have
N (t) ∼ Poisson(λ = αt)
Examples...
Summary of Today’s Class
• We introduced the Poisson distribution and
Poisson process.
Homework:
• Problem set 12