Transcript Chapter 17 Probability Models - Washington University in

Chapter 17 Probability Models
math2200
I don’t care about my [free throw shooting] percentages. I
keep telling everyone that I make them when they count.
-- Shaquille O'Neal, in post-game interviews recorded by
WOAI-TV on November 7, 2003.
O’Neal’s free throws
• Suppose Shaq shoots 45.1% on average .
Let X be the number of free throws Shaq
needs to shoot until he makes one.
Pr (X=2)=? Pr (X=5)=? E(X)=?
Bernoulli trials
• Only two possible outcomes
– Success or failure
• Probability of success, denoted by p the
same for every trial
• The trials are independent
• Examples
– tossing a coin
– Free throw in a basketball game assuming
every time the player starts all over.
Can we model drawing without
replacement by Bernoulli trials?
• The draws are not independent when
sampling without replacement in finite
population. But they are be treated as
independent if the population is large
– Rule of thumb: the sample size is smaller than
10% of the population
Geometric model
• How long does it take to achieve a success in
Bernoulli trials?
• A Geometric probability model tells us the
probability for a random variable that counts the
number of Bernoulli trials until the first success
• Geom(p)
–
–
–
–
–
–
p = probability of success
q = 1-p = probability of failure
X : number of trials until the first success occurs
P (X=x) = qx-1 p
E (X) = 1/p
Var (X) = q/p2
• What is the probability that Shaq makes
his first free throw in the first four attempts?
• 1-P(NNNN) = 1-(1-0.451)4 = 0.9092
or P(X=1)+P(X=2)+P(X=3)+P(X=4)
Binomial model
• A Binomial model tells us the probability
for a random variable that counts the
number of successes in a fixed number of
Bernoulli trials.
The Binomial Model (cont.)
• There are
n!
n Ck 
k ! n  k !
ways to have k successes in n trials.
– Read nCk as “n choose k.”
The Binomial Model (cont.)
• Binom (n, p)
n = number of trials
p = probability of success
q = 1 – p = probability of failure
X = number of successes in n trials
n!
 n  x n x
n
P( X  x)    p q where  
 x
 x  x !(n  x)!
  np
  npq
How do we find E(X) and Var(X)?
• Find P(X=x) directly
• Binomial random variable can be viewed as the
sum of the outcome of n Bernoulli trials
– Let Y1,…, Yn be the outcomes of n Bernoulli trials
– E (Y1) =…= E (Yn) = p*1+q*0=p,
E(X) = np
– Var (Y1) =…= Var (Yn) = (1-p)2 *p+(0-p)2 *q = pq,
Var (X) = npq.
• In general if Y1,…,Yn are independent and have
the same mean µ and variance σ2 and X =
Y1+…+Yn, then E(X) = E(Y1)+…+E(Yn)=nµ
and Var(X) = Var(Y1)+…+Var(Yn)=nσ2 .
• If Shaq shoots 20 free throws, what is the
probability that he makes no more than
two?
• Binom(n, p), p=0.451, n=20
P(X=0 or 1 or 2) = P(X=0) + P(X=1) +
P(X=2) = 0.0009
Normal approximation to Binomial
• If X ~ Binomial (n, p), n =10000, p =0.451,
P(X<2000)=?
• When Success/failure condition (np >= 10
and nq>=10) is satisfied, Binomial (n,p)
can be approximated by Normal with
mean np and variance npq.
• P (X<2000)=P ( Z< (2000-np) / sqrt (npq))
= P(Z< -50.4428)
=normalcdf (-1E99, -50.4428 ,0,1) = 0
Poisson model
• Binomial(n,p) is approximated by Poisson(np) if
np<10.
• Let λ=np, we can use Poisson model to
approximate the probability.
• Poisson(λ)
– λ : mean number of occurrences
– X: number of occurrences

e 
P  X  x 
x!
EX   
x
SD  X   
Poisson Model (cont.)
• The Poisson model is also used directly to model the
probability of the occurrence of events.
• It scales to the sample size
– The average occurrence in a sample of size 35,000 is
3.85
– The average occurrence in a sample of size 3,500 is
0.385
• Occurrence of the past events doesn’t change the
probability of future events.
An application of Poisson model
• In 1946, the British
statistician R.D.
Clarke studied the
distribution of hits of
flying bombs in
London during
World War II.
• Were targeted or
due to chance.
Flying bomb (cont’)
• The average number of hits per square is then
537/576=.9323 hits per square. Given the number of hits
following a Poisson Model
P (X=0) = [e^(-0.923)*(-0.923)^0] / 0! = 0.393647
0.393647* 576 = 226.7
# of hits
0
1
2
3
4
5
# of cells with # of hits above
229
211
93
35
7
1
Poisson Fit
226.7
211.4
98.5
30.6
7.1
1.6
• No need to move people from one sector to another,
even after several hits!
What Can Go Wrong?
• Be sure you have Bernoulli trials.
– You need two outcomes per trial, a constant
probability of success, and independence.
– Remember that the 10% Condition provides a
reasonable substitute for independence.
• Don’t confuse Geometric and Binomial
models.
• Don’t use the Normal approximation with
small n.
– You need at least 10 successes and 10
failures to use the Normal approximation.
What have we learned?
– Geometric model
• When we’re interested in the number of Bernoulli trials until
the first success.
– Binomial model
• When we’re interested in the number of successes in a fixed
number of Bernoulli trials.
– Normal model
• To approximate a Binomial model when we expect at least 10
successes and 10 failures.
– Poisson model
• To approximate a Binomial model when the probability of
success, p, is very small and the number of trials, n, is very
large.
TI-83
• 2nd + VARS (DISTR)
• pdf: P(X=x) when X is a discrete r.v.
– geometpdf(prob,x)
– binompdf(n,prob,x)
– poissonpdf(mean,x)
• cdf: P(X<=x)
– geometcdf(prob,x)
– binomcdf(n,prob,x)
– poissoncdf(mean,x)