It Can Be Stated That If the R

1. Bernoulli Distribution

For an experiment in which there are only 2 possible outcomes (e.g. Head –tail, defective – nondefective, success – failure etc.)
usually denoted by 0, 1

A random variable x has a Bernoulli distribution with parameter p (0 ≤ p ≤ 1) if x can take on only the values o and 1, and the probabilities are :
P (x = 1) = p Prob. of success
P (x = 0) = 1 – p = q Prob. of failure
px q1-x for x = 0 , 1
f(x ; p) =
0 otherwise

Prob. Mass Fnc. (pmf) of x p

(Discrete density) q

0 1 x

meanE(x) = = 0.q + 1.p= p

E (X) = p

Variance = E(x - µ)2

= E(x 2) – [E(x)]2 ------(1)

but E(x2) = = 12. p + 02. q = p

From (1) Variance = p – p2 = p(1 – p) = pq

Var (X) = pq

2. Binomial Distribution

x has a binomial distribution when :

The experiment consists of n Bernoulli trials

(that result in either a “success” (s) or a “failure” (f)

The trials are identical and independent and therefore the probability of success, p, remains the same from trial to trial (constant probability)
The random variable x denotes the number of

successes obtained in the n trials,

Def. Binomial Distribution
A random variable X has a binomial distribution w / parameters n and p if its density is given by
, x = 0 , 1 , 2 , … n
b (x, n, p) = 0 otherwise

It can be stated that if the random variable X1,…,Xn form n Bernoulli trials with parameters p, and if X = X1+….+ X n then x has a binomial distribution with parameters n and p.

Ex.11 It has been claimed that in 60 % of all solar heat installation the utility bill is reduced by at least one – third. Accordingly, what are the probabilities that the utility bill will be reduced by at least one-third in

(a)four of five installation;

(b)at least four of five installation?

The mean and variance of X

If X~Bin(n, p) , then
E(X) = np
V(X) = np(1-p) = npq , where q = 1-p
 =

Ex.12 Rubber bands are produced with a designed breaking strength. Mr.Reeve took a sample of 50 from the manufacturing line, stretched them to the designed strength, and 3 bands broke.

(a)What is the prob. of this observation?

(b)What is the prob. that more than 3 rubber bands will break in a sample of 50 ?

(c)How many are expected to break in a test of 100 bands?

______

Probability Mass Function

b (X; n, p) = , where X = 0, 1, …, n

and 0<p<1

b(x) p = q = 0.5 เช่น b (x; 8, 0.5)

symmetric

b(x)p < 0.5 เช่น b(x; 8, 0.25)

skewed right

b(x)p > 0.5 เช่น b(x; 8, 0.75)

skewed left

Cumulative Distribution Function (CDF)

B(x) = p(X x) = B (x; n, p) =

โดยx = 0, 1,.., n

Note (1) p (X) = b (X; n, p)

= B (X; n, p) – B(X-1, n, p)

(2) = 1 - B (X-1, n, p)

Binomial Tables - cumulative distribution

B(x; n, p)

Ex. 13 Suppose that 20 % of all copies of a particular textbook fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test. Then X has a binomial distribution with n = 15, p=0.2

(a)Find the prob. that at most 8 fail the test.

(b)Find the prob. that exactly 8 fail.

3. Multinomial Distribution

Used in experiment that can result in more than 2 types of outcomes, e.g. picking a card from the deck by replacing each one before the next card is pulled out. In this case, the outcome can occur in any of the 4 ways: spade, heart, diamond, and flower.

Consider the general case, if the outcomes of each experiment can result in k ways: E1, E2, …….Ek with probability p1, p2, …….pk, respectively.

Multinomial Distribution gives the probability of performing n experiments, at which eventE1 occurs x1 times, E2 occurs x2 times,.... Ekoccurs xk times,where

and

Prob. Of getting event E1 , E2, … Ek
f (x1, x2,…, xk; p1, p2,…, pk, n) =

Proof 1. Since each trial is independent of one another, in n trials theprobability ofE1occurringx1 times, E2occurringx2 times, .... Ekoccurringxktimes (in a particular sequence) is

2. Find the number of ways in which arrangements can be made. This is equivalent to

3. Since each ordered arrangement is mutually exclusive, the probability of E1 occurring x1 times, E2occurring x2 times,.... Ekoccurring xk timesin n trials is

เทอม

Ex.14 Roll 2 dies for 7 times. Find the probability of the result consisting of the following events: total score of 7occurring 3 times, total score of 11 occurring 1 time, same score for both dies occurring 2 times, and event other than the above occurring 1 time.

Sol. LetE1 = total score of 7

= (1, 6) (2, 5), (4, 3), (6, 1), (5, 2), (3, 4)

E2 = total score of 11

= (5, 6), (6, 5)

E3 = same score for both dies

= (1, 1),…(6, 6)

E4 = event other than the above

4. Hypergeometric Distribution

There are 2 types of sampling:sampling with replacement,sampling without replacement

1)Sampling with replacement - Eachunit drawn from the sample was returned to the lot before the next sampling process. (In this case, the probability of success for each trial is constant, and each trial is independent of one another.

In this case, we use Binomial Distribution orMultinomial Distribution

2) Sampling without replacement –Each unit drawn from the sample is not put back to the lot before the next sampling process. (In this case, the probability of success for each trial is not constant, and each trial is not independent of one another).

In this case, we useHypergeometric Distribution.

Suppose that we have a group of N object, of which k are defective. We want to take a sample of n objects from the group without replacement. Let x denote the number of defectives obtained from the sample, then x has a hypergeometric distribution.

Def. Hypergeometric Distribution
A random variable x has a hypergeometric distribution with parameters N, n, and K if its density is given by
, for x=0, 1, ….n
= probability of getting x successes

Proof

1. The number of ways of choosing n subjects from a population of size N = ways

2. The number of ways in which x successes are chosen from k successes=

3. The number of ways in which n-x failures are chosen from N-k failures =

4. Therefore, the number of ways in which x successes and n-x failures are chosen is

 The probability of getting x successes in n trials is

Mean correction here

Variance

Ex. 15 A shipment of 20 tape recorders contains 5 that are defective. If 10 of them are randomly chosen for inspection, what is the prob. that 2 of the 10 will be defective?

N= 20, k= 5, n=10, x=2

h =

Ex.16 A special close-tolerance retaining ring was manufactured for the space shuttle spacecraft. The machine which produced the ring has a defective rate of 8%. If 25 rings were made, determine (a) the chance of observing less than 2 defectives in a sample of 8 and (b)  and of the number of defectives in samples of size 8.

Ex.17 A foundry ships engine blocks in lots of size 20. Since no manufacturing process is perfect, defective blocks are inevitable. However, to detect the defect, the block must be destroyed. Thus we cannot test each block. Before accepting a lot, 3 items are selected and tested. Suppose that a given lot actually contains five defective items. What is the expected number of defective blocks in the sample selection.

Binomial Approximation to the Hypergeometric Distribution

For hypergeometric experiment, if the sample size n is very small compared to the population size N, each experiment almost has no effect on the other experiments, meaning that sampling with replacement yields very close results to sampling without replacement.

Therefore, we can represent hypergeometric distribution by binomial distribution with

when n is very small compared with N (if)that is,

h (x, n, k, N) ~ b(x; n, p) where
if
Mean ofbinomial  = np =
2of Binomial 2 = npq=

Note

is similar to the formula for ofhypergeometric distribution, but it is different in the term which is close to 1 when n is very small compared to N

Ex.18 During an hour, 1000 bottles of beer are filled by a machine. Each hour a sample of 20 bottles is randomly selected and the number of ounces of beer per bottle is checked. Suppose that during a particular hour, 100 underfilled bottles are produced. Find the prob. that at least 3 underfilled bottles will be among those sampled.

Ex.19In choosing 20 students randomly from a university which has 4900 males and 2100 females to participate in an opinion survey, find the probability that 12 male students are selected.

5. Poisson Distribution

Poisson experiment is an experiment in which the variable X represents the number of successes in a particular time, or space, or volume.

Ex. X= number of times of incoming telephones to the company

X = number of school holidays in a semester

X = number of rats in 1-acre land

X = number of bacteria in the Petri-dish

X = number of mistakes in a page

Characteristics of Poisson Process

1. There are n independent trials where n is very large.

2. Only 1 outcome is of interest on each trial.

3. There is a constant probability of occurrence on each trial.

Def. A random variable X is said to have a Poisson Distribution with parameter k if its density f is given by f (x;) = , > 0, X= 0, 1, 2,….
f (x; k) = X= 0, 1, 2,….
k > 0
Mean E (x) =
Variance Var x =

Notation Letα= expected number of occurrences per - unit time

- unit length

- unit space

 The number of occurrences during timet will havepoisson distribution withmean of = αt

f (x; αt) =

Ex.20 Suppose that radioactive particles strike a target in accordance with a Poisson process at an average rate of 3 particles per minute.

a) Determine the prob. that 10 particles will strike the target in a 2-minute period.

b) Determine the prob. that 10 or more particles will strike the target in a 2-minute period.

Ex.21 The white blood cell count of a healthy individual can average as low as 6000/m3 of blood. To detect a white – cell deficiency, a 0.001 m3 drop of blood is taken and the number of white cells X is found. How many white cells are expected in a healthy individual?

Ex.22 From the survey of accidents on a particular road, the average number of accidents is 4 times a week. Find the probability of accidents occurring 6 times in a week.

Ex.23 In counting the number of bacteria on the microscope, it is found that the average number of bacteria on the lens is 15. Determine the probability of finding 20 bacteria on the lens.

Ex.24 The average number of airplanes landing at an airport is 5 planes/ hour. Determine the probability of having 10 planes landing at this airport in a particular hour.

Poisson Approximation to the Binomial Distribution

Let X bea binomial random variable with probability distribution b(x; n, p). If, (n≥100, np≤10)

Binomial Distribution can be approximate byPoisson Distribution f(x; k) with = np

b (x; n, p) f (x; ) =

Ex.25 In manufacturing electronic circuits, ceramic plates are drilled to provide path ways called ‘vias’ from one surface to another. A typical plate is about the size of a playing card and may require 10,000 vias each as small as a pinpoint. In the past these vias were drilled using diamond drills. New technology uses lasers to produce these precisely positioned pathways. Suppose that the probability of incorrect positioning a via is only 1/20,000 What is the probability that a randomly selected plate will have no improperly positioned vias?

Ex.26 2% of the screws made by a machine are defective, the defectives occurring at random during production. If the screws are packaged 100 per box, what is the probability that a box contains x defectives?

Ex.27 A light bulb factory found 3% defectives in a day’s inspection. If 100 bulbs are randomly selected, find the probability that 5 defectives are found.

P(x = 5)

6. Negative Binomial Distribution

(Pascal Distribution)

Consider an experiment similar to binomial experiment, but instead of having x successes in n trials, we look for the probability of having kth success in a total of x trials. This is called “Negative Binomial Experiment” b* (x; k, p)

Definition Random Variable X that represents the number of trials which results in kth success in a negative binomial experiment is calledNegative binomial random variable.

p = probability of success

X = the total number of trials that result of kth success

Formula forb* (x; k, p)

1. Find the probability of having kth success after having k-1 successes earlier, and failure x – k, in x trials

Let p = probability of success

q = 1 - p = prob. of failure

Since each trial is independent of one another, the probability of an event having a particular sequence is pk-1 qx- kp = pk qx-k

2. Find the number of ways that the trials end with success after having k-1 successes and x-k failures.

3. So the probability of trial that ends with success after having k -1 successes and x-k failures is:

pk q x-k + pk q x-k+ … + pk q x-k

terms

b* (x; k, p) = pk qx-k
wherex= k, k+1,…
mean  =
variance

Ex.28 Toss two coins for 5 times. Determine the probability of getting both heads for the 3rd time on the 5th toss.

X = 5, k = 3, p = ½ *1/2 = ¼

Note The reason this is called “negative binomial”is because b*(x; k, p) wherex= k, k+1,… is the terms derived frombinomial expansion ofpk(1-q)-k

Consider the case of k=1

(i.e. determine the probability of the number of times that the experiment ends with success for the first time).

b* (x; 1, p) = pqx-1
,where x= 1, 2, 3,….

Sinceb*(x; 1, p) is the xth term inGeometric Progression, we called this“Geometric Distribution” (Negative Binomial Distribution atk =1)

7. Geometric Distribution

If the experiment consists of several trials which are independent of one another, and each has probability of success = p, and probability of failure= q

And if xis arandom variable that represents the number of times that the trial ends with success for the 1st time, then theprobability distribution ofXis:

g (x, p) = pqx-1 wherex = 1, 2, 3,…
 =

Ex.29 A scientist wants to cultivate bacteria in several rats by injecting the culture into each rat. After each injection he checks whether the rat is infected. If it is, he stops doing the experiment.If the probability that each rat is infected is 1/5, determine the probability that the scientist uses only 6 rats in this experiment.

Geometric, X = 6, p = 1/5

Ex.30 A grocery product company markets 175 different products. Data indicates that there is a 20% chance that in any one year, legal proceedings will be initiated by a consumer or distributor against the company concerning any one of these products. A new product is to be introduced.

(a)Compute the probability of the first lawsuit in the third year of marketing

(b)Compute  and  for the distribution.

Midterm – Ch 3 binomial, poisson, geometric, hypergeometric, Uniform