multinomial experiments

JMB Ch. 5 Part 2 rev 2012b EGR 252.001 2012 9th ed. Slide 1

Multinomial ExperimentsWhat if there are more than 2 possible outcomes?

(e.g., acceptable, scrap, rework)That is, suppose we have:

n independent trialsk outcomes that are

mutually exclusive (e.g., ♠, ♣, ♥, ♦) exhaustive (i.e., ∑all kpi = 1)

Then f(x1, x2, …, xk; p1, p2, …, pk, n) =

kxk

xx

k

pppxxx

n...

,...,,21

2121


Multinomial Examples Example 5.7 refer to page 150

Problem 22, page 152 Convert ratio 8:4:4 to probabilities (8/16, 4/16)

f( __, __, __; ___, ___, ___, __) =___.25, 8) = (8 choose 5,2,1)(0.5)5(0.25)2(0.25)1 = 8!/(5!2!1!)* )(0.5)5(0.25)2(0.25)1 = 21/256 or 0.082031

x1 = _______ p1 = 0.50

x2 = _______ p2 = 0.25

x3 = _______ p3 = 0.25


Binomial vs.Hypergeometric Distribution

Replacement and IndependenceBinomial (assumes sampling “with replacement”)

and hypergeometric (sampling “without replacement”)

Binomial assumes independence, while hypergeometric does not.

Hypergeometric: The probability associated with getting x successes in the sample (given k successes in the lot.)

n

N

xn

kN

x

k

knNxhxXP ),,;()(


Example from Complete Business Statistics, 4th ed (McGraw-Hill) Automobiles arrive in a dealership in lots of 10. Five out of

each 10 are inspected. For one lot, it is known that 2 out of 10 do not meet prescribed safety standards. What is probability that at least 1 out of the 5 tested from that lot will be found not meeting safety standards?

This example follows a hypergeometric distribution: A random sample of size n is selected without replacement

from N items. k of the N items may be classified as “successes” and N-k are

“failures.” The probability associated with getting x successes in the sample

(given k successes in the lot.)

n

N

xn

kN

x

k

knNxhxXP ),,;()(

Hypergeometric Example


Solution: Hypergeometric Example In our example,

k = number of “successes” = 2 n = number in sample = 5

N = the lot size = 10 x = number found = 1 or 2

P(X > 1) = 0.556 + 0.222 = 0.778

5

10

25

210

2

2

5

10

15

210

1

2

)2,5,10;2()2,5,10;1(

)2()1()(

hh

XPXPxXP


Expectations: Hypergeometric Distribution

The mean and variance of the hypergeometric distribution are given by

What are the expected number of cars that fail inspection in our example? What is the standard deviation?

μ = nk/N = 5*2/10 = 1

σ2 = (5/9)(5*2/10)(1-2/10) = 0.444

σ = 0.667

N

k

N

kn

N

nNN

nk

1**1

2


Additional problems …A worn machine tool produced defective parts for a period of time before the problem was discovered. Normal sampling of each lot of 20 parts involves testing 6 parts and rejecting the lot if 2 or more are defective. If a lot from the worn tool contains 3 defective parts:

1. What is the expected number of defective parts in a sample of six from the lot? N = 20 n = 6 k = 3 μ = nk/N = 6*3/20 =18/20=0.9

2. What is the expected variance? σ2

= (14/19)(6*3/20)(1-3/20) = 0.5637

3. What is the probability that the lot will be rejected?

P(X>2) = 1 – [P(0)+P(1)]


Binomial Approximation Note, if N >> n, then we can approximate the

hypergeometric with the binomial distribution. Example: Automobiles arrive in a dealership in lots of 100. 5

out of each 100 are inspected. 2 /10 (p=0.2) are indeed below safety standards. What is probability that at least 1 out of 5 inspected will not meet safety standards?

Recall: P(X ≥ 1) = 1 – P(X < 1) = 1 – P(X = 0)

Hypergeometric distribution Binomial distribution

1 - h(0;100,5,20)

= 0.6807

1 - b(0;5,0.2)

1 - 0.3277 = 0.6723

(See also example 5.12, pg. 155-6)


Negative Binomial Distribution b* A binomial experiment in which trials are

repeated until a fixed number of successes occur.

Example:Historical data indicates that 30% of all bits transmitted through a digital transmission channel are received in error. An engineer is running an experiment to try to classify these errors, and will start by gathering data on the first 10 errors encountered.

What is the probability that the 10th error will occur on the 25th trial?


This example follows a negative binomial distribution: Repeated independent trials. Probability of success = p and probability of failure = q = 1-p. Random variable, X, is the number of the trial on which the kth

success occurs. The probability associated with the kth success occurring

on trial x is given by,

Where,k = “success number”x = trial number on which k occurs p = probability of success (error) q = 1 – p

,...2,1,,1

1),;(*

kkkxqpk

xpkxb kxk

Negative Binomial Equation


Example: Negative Binomial DistributionWhat is the probability that the 10th error will

occur on the 25th trial? k = “success number” = 10

x = trial number on which k occurs = 25

p = probability of success (error) = 0.3

q = 1 – p = 0.7

102510 )7.0()3.0(110

125)3.0,10;25(*

b

037.0)7.0()3.0(9

24)3.0,10;25(* 1510

b


Geometric DistributionContinuing with our example in which p =

probability of success (error) = 0.3What is the probability that the 1st bit received in

error will occur on the 5th trial?This is an example of the geometric distribution,

which is a special case of the negative binomial in which k = 1.The probability associated with the 1st success

occurring on trial x is

P = (0.3)(0.7)4 = 0.072

...,3,2,1);( 1 xpqpxg x


Additional problems …A worn machine tool produces 1% defective parts. If we assume that parts produced are independent:

1.What is the probability that the 2nd defective part will be the 6th one produced?

2.What is the probability that the 1st defective part will be seen before 3 are produced?

3.How many parts can we expect to produce before we see the 1st defective part?

Negative binomial or geometric? Expected value = ?


Poisson Process

The number of occurrences in a given interval or region with the following properties: “memoryless” ie number in one interval is independent of

the number in a different intervalP(occurrence) during a very short interval or small region

is proportional to the size of the interval and doesn’t depend on number occurring outside the region or interval.

P(X>1) in a very short interval is negligible


Poisson Process Situations

Number of bits transmitted per minute.Number of calls to customer service in an hour.Number of bacteria present in a given sample.Number of hurricanes per year in a given region.


Poisson Distribution ProbabilitiesThe probability associated with the number of

occurrences in a given period of time is given by,

Where,λ = average number of outcomes per unit time or region

t = time interval or region

...,2,1,0,!

)();(

xx

tetxp

xt


Service Call Example - Poisson Process

Example

An average of 2.7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. To determine personnel and equipment needs to maintain a desired level of service, the plant manager needs to be able to determine the probabilities associated with numbers of service calls.


Our Example: λ = 2.7 and t = 1 minute

What is the probability that fewer than 2 calls will be received in any given minute?

The probability that fewer than 2 calls will be received in any given minute is

P(X < 2) = P(X = 0) + P(X = 1)

The mean and variance are both λt, so

μ = λt =________________

Note: Table A.2, pp. 732-734, gives Σt p(x;μ)

!1

)7.2(

!0

)7.2()2(

17.207.2

ee

XP


Service Call Example (Part 2) If more than 6 calls are received in a 3-minute period,

an extra service technician will be needed to maintain the desired level of service. What is the probability of that happening? μ = λt = (2.7) (3)= 8.1 8.1 is not in the table; we must use basic equation

Suppose λt = 8; see table with μ = 8 and r = 6

P(X > 6) = 1 – P(X < 6) = 1 - 0.3134 = 0.6866

!6

)1.8(

!1

)1.8(

!0

)1.8(1)6(1)6(

61.811.801.8 eeeXpXp


Poisson Distribution

05

101520253035404550

Fre

qu

ency

Calls per minute

multinomial experiments

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