Download Notes 12 - Wharton Statistics

Statistics 510: Notes 12
Reading: Sections 4.8-4.9
I. Review: Poisson Distribution
Arises in two settings:
(1) Poisson distribution provides an approximation to the
binomial distribution when n is large, p is small and
  np is moderate.
(2) Poisson distribution is used to model the number of
events that occur in a time period t when
(a) the probability of an event occurring in a given small
time period h is approximately proportional to  h .
(b) the probability of two or more events occurring in a
given small time period h is much smaller than  h .
(c) the number of events occurring in two non-overlapping
time periods are independent
When (a), (b) and (c) are satisfied, the number of events
occurring in a time period t has a Poisson ( t ) distribution.
The parameter  is called the rate of the Poisson
distribution;  is the mean number of events that occur in
a time period of length 1. The mean number of events that
occur in a time period of length t is t and the variance of
the number of events is also t .
Sketch of proof for Poisson distribution under (a)-(c):
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For a large value of n, we can divide the time period t into
n nonoverlapping intervals of length t / n . The number of
events occurring in time period t is then approximately
Binomial ( n,  t / n) . Using the Poisson approximation to
the binomial, the number of events occurring in time period
t is approximately Poisson (n * t / n) =Poisson ( t ).
Taking the limit as n   yields the result.
Number of events occurring in space: The Poisson
distribution also applies to the number of events occurring
in space. Instead of intervals of length t, we have domains
of area or volume t. Assumptions (a)-(c) become:
(a’) the probability of an event occurring in a given small
region of area or volume h is approximately proportional to
h .
(b’) the probability of two or more events occurring in a
given small region of area or volume h is much smaller than
h
(c’) the number of events occurring in two non-overlapping
regions are independent
The parameter  for a Poisson distribution for the number
of events occurring in space is called the intensity.
Example 1: During World War II, London was heavily
bombed by V-2 guided ballistic rockets. These rockets,
luckily, were not particularly accurate at hitting targets.
The number of direct hits in the southern section of London
has been analyzed by splitting the area up into 576 sectors
measuring one quarter of a square kilometer each. The
average number of direct hits per sector was 0.9323. The
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fit of a Poisson distribution with   0.9323 to the observed
frequencies is excellent:
Hits
Actual Frequency Expected
Frequency
(576*P(X=k hits))
where
X~Poisson(0.9323)
0
229
226.74
1
211
211.39
2
93
98.54
3
35
30.62
4
7
7.14
5 or more
1
1.57
R Commands for Poisson distribution:
The command rpois(n,lambda) simulates n Poisson random
variables with parameter lambda.
The command dpois(x,lambda) computes the probability
that a Poisson random variable with parameter lambda
equals x.
The command ppois(x,lambda) computes the probability
that a Poisson random variable with parameter lambda is
less than or equal to x.
II. Geometric Random Variable (Section 4.8.1)
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Suppose that independent trials, each having a probability
p, 0  p  1 , of being a success, are performed until a
success occurs. Let X be the random variable that denotes
the number of trials required. The probability mass
function of X is
P{ X  n}  (1  p)n1 p
n  1, 2,
(1.1)
The pmf follows because in order for X to equal n, it is
necessary and sufficient that the first n-1 trials are failures
and the nth trial is a success.
A random variable that has the pmf (1.1) is called a
geometric random variable with parameter p.
The expected value and variance of a geometric (p) random
variable are
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1 p
E ( X )  , Var ( X )  2 .
p
p
Example 2: A fair die is tossed. What is the probability
that the first six occurs on the fourth roll? What is the
expected number of tosses needed to toss the first six?
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III. Negative Binomial Distribution (Section 4.8.2)
Suppose that independent trials, each having a probability
p, 0  p  1 , of being a success, are performed until r
successes occur. Let X be the random variable that denotes
the number of trials required. The probability mass
function of X is
 n  1 r
n r
P{ X  n}  
n  r , r  1, (1.2)
 p (1  p)
 r 1 
A random variable whose pmf is given by (1.2) is called a
negative binomial random variable with parameters ( r , p ) .
Note that the geometric random variable is a negative
binomial random variable with parameters (1, p) .
The expected value and variance of a negative binomial
random variable are
r
r (1  p)
E ( X )  , Var ( X ) 
p
p2
Example 3: Suppose that an underground military
installation is fortified to the extent that it can withstand up
to four direct hits from air-to-surface missiles and still
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function. Enemy aircraft can score direct hits with these
particular missiles with probability 0.7. Assume all firings
are independent. What is the probability that a plane will
require fewer than 8 shots to destroy the installation? What
is the expected number of shots required to destroy the
installation?
IV. Hypergeometric Random Variables (Section 4.8.3)
Suppose that a sample of size n is to be chosen randomly
(without replacement) from an urn containing N balls, of
which m are white and N  m are black. If we let X be the
random variable that denotes the number of white balls
selected, then
 m  N  m 
 

i
n

i
 , i  0,1, , n
P{ X  i}   
N
(1.3)
 
n 
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A random variable X whose pmf is given by (1.3) is said to
be a hypergeometric random variable with parameters
n, N , m .
The expected value and variance of a hypergeometric
random variable with parameters n, N , m are
nm
n 1 

E( X ) 
, Var ( X )  np (1  p ) 1 

N
 N 1  .
Example 4: A Scrabble set consists of 54 consonants and
44 vowels. What is the probability that your initial draw
(of seven letters) will be all consonants? six consonants
and one vowel? five consonants and two vowels?
V. Zeta (or Zipf) distribution
A random variable is said to have a zeta (sometimes called
the Zipf) distribution with parameter  if its probability
mass function is given by
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C
, k  1, 2,
k  1
for some value of   0 .
P{ X  k} 
Since the sum of the foregoing probabilities must equal 1, it
follows that
   1  1 
C     
 k 1  k  
1
The zeta distribution has been used to model the
distribution of family incomes.
VI. The Cumulative Distribution Function (Section 4.9)
The cumulative distribution function (CDF) of a random
variable X is the function F (b)  P( X  b) .
Example 5: Let X denote the number of aces a poker player
receives in a five card hand. Graph the cdf of X.
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All probability questions about X can be answered in terms
of the cdf F. For example,
P(a  X  b)  F (b)  F (a ) for all a  b .
This can be seen by writing the event { X  b} as the union
of the mutually exclusive events { X  a} and {a  X  b} .
That is,
{ X  b}  { X  a}  {a  X  b} so
P{ X  b}  P{ X  a}  P{a  X  b} .
The probability that X  b can be computed as
1
P( X  b)  P(lim n { X  b  })
n
1
 lim n P( X  b  )
n
1
 lim n F (b  )
n
For the justification of the second equality, see Section 2.6
on the continuity property of probability.
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