# ch05

Pages: 8 (1166 words) Published: July 9, 2015
Distribution Families
• Named family of distributions characterized by a single
pdf with one or more parameters.
– Normal distribution  parameters: , 

• The different distribution families often have theoretical reasons for being used in different practical cases:
– Normal: measurement & dimension data
– Lognormal: manual service & repair times
– Poisson: # of events occurring in a time period, distance, or area
– Exponential, Weibull: Time to failure of equipment

Ch. 5: Discrete Distributions
1. Uniform
2. Binomial
3. Hypergeometric
4. Negative Binomial
5. Geometric
6. Poisson
SKIPPING: Multinomial (p/149-150)

Discrete Uniform Distribution

Bernoulli Process

Binomial Distribution

f(x;n,p)=

=average number of successes in n trials

Binomial Tables (in text)

Problem
• The probability that a patient recovers from a
delicate heart operation is 0.9. What is the
probability that exactly 5 of the next 7 patients
having this operation survive?

Negative Binomial Distribution

k

p
k (1  p)
 
p2
2

Note: Not in
textbook

Problem
The probability that a patient recovers from a delicate
heart operation is 0.9.
• What is the probability that the 7th patient is the 5th
patient to survive the operation?
• On average, how many patients will undergo the
surgery before a patient does not survive?

Binomial vs. Negative Binomial
Binomial

Negative Binomial

• Based on Bernoulli process
• Probability of x successes in
n trials

• Based on Bernoulli process
• Probability of trial x being
the kth success

• Focus: # successes

• Focus: # trials

 n
b( x : n, p)    p x q n  x
 x

 x  1
k xk
b *( x : k , p)  
p
 q
 k  1

Geometric Distribution
• Special case of negative binomial
• Focus: finding the trial of the 1st success
• Example: “What is the chance that it will take
5 trials before a success occurs?”

Geometric Distribution

Example
Suppose the probability is 0.8 that any given person will
believe a tale about the transgressions of a famous
actress. What is the probability that
(a) The third person to hear this tale is the first one to
believe it?
(b) The sixth person to hear this tale is the fourth one to
believe it?
(c) On average, how many people have to hear the story
until someone believes it?
(d) Four of the next 7 people believe the tale?

Hypergeometric
• rv
X = # of successes in a random sample of
size n selected from N items of which k are
labeled successes and N-k labeled failures.
• Application: acceptance sampling
• pmf:
for max(0,n-(N-k))x min(n,k)

Hypergeometric
• Example. Truck contains 40 parts of which 13
are defective. A sample of 10 is drawn. Let X
be number of defects in sample.
Pr(X=2)?
E[X]?

Properties of Poisson Processes
1. Memory-less
2. Probability of a single outcome in a small
space/interval is proportional to the size of
the space/interval
3. Probability of >1 outcome in a small space is
negligible (zero)

Poisson Distribution

Poisson Table 2

Poisson Example
• A certain area of the eastern US is, on average,
hit by 6 hurricanes a year. Find the probability
that for a given year that area will be hit by
a) Fewer than 4 hurricanes
b) Anywhere from 6 to 8 hurricanes

Binomial  Poisson
• A binomial random variable X can be
approximated with a Poisson distribution when
– n is very large (at least 20)
– p is very small (less than 0.05)
– Approximation improves as n gets larger and p smaller

• Use the relationship
 = np = t
Can then use Poisson tables to solve problem

Binomial  Poisson Example
• Suppose that, on average, 1 person in 1000
makes a numerical error in preparing his or
her income tax return. If 10,000 forms are
selected at random and examined, find the
probability that 6, 7, or 8 of the forms contain
an error.

Using the Tables - Problems

B(0; 7, 0.4)?...

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