Chapter 6 Some Special Discrete Chapter 6 Some Special Discrete DistributionsDistributions

6.1 THE BERNOULLI DISTRIBUTION6.1 THE BERNOULLI DISTRIBUTION

6.2 THE BINOMIAL DISTRIBUTION6.2 THE BINOMIAL DISTRIBUTION

6.3 THE GEOMETRIC DISTRIBUTION6.3 THE GEOMETRIC DISTRIBUTION

6.4 THE POISSON DISTRIBUTION6.4 THE POISSON DISTRIBUTION

6.1 THE BERNOULLI 6.1 THE BERNOULLI DISTRIBUTIONDISTRIBUTION

6.1.1 6.1.1 Bernoulli Trials Bernoulli Trials

Example 1Example 1– The calculation of a payroll check may be correct or incoThe calculation of a payroll check may be correct or inco

rrect. We define the Bernoulli random variable for this trirrect. We define the Bernoulli random variable for this trial so that X = 0 corresponds to a correctly calculated chal so that X = 0 corresponds to a correctly calculated check and X = 1 to an incorrectly calculated one.eck and X = 1 to an incorrectly calculated one.

Example 2Example 2– A consumer either recalls the sponsor of a T.V. program A consumer either recalls the sponsor of a T.V. program

(X = 1) or does not recall (X = 0)(X = 1) or does not recall (X = 0)

Example 3Example 3– In a process for manufacturing spoons each spoon may In a process for manufacturing spoons each spoon may

either be defective(X = 1) or not (X = 0). either be defective(X = 1) or not (X = 0).

And, the probability distribution is And, the probability distribution is

X =

0

1.

,

failureafor

successaforX =

xp

XX 1 0

P(X = x) pp 1-p

Mathematically, the Bernoulli distribution is given by Mathematically, the Bernoulli distribution is given by

To calculate the mean and variance,To calculate the mean and variance,

Mean, Mean, = E(X) = = E(X) =

Variance, Variance, 22 = E(X = E(X22) - ) - = =

Note: Since Bernoulli distribution is determined by the value of p, p is tNote: Since Bernoulli distribution is determined by the value of p, p is the he parameterparameter of this distribution of this distribution

P(X = x) = px ( 1- p)1 - x for x = 1,0

1

0

)(x

xXPx

1

0

2 )(x

xXPx

1

0

2 )(x

xXPx

6.2 THE BINOMIAL DISTRIBUTION6.2 THE BINOMIAL DISTRIBUTION

6.2.1 The Probability Functions6.2.1 The Probability Functions– Consider an experiment which has two possible outcomConsider an experiment which has two possible outcom

es, one which may be termed ‘success’ and the other ‘faes, one which may be termed ‘success’ and the other ‘failure’. A binomial situation arises when n independent triilure’. A binomial situation arises when n independent trials of the experiment are performed , for exampleals of the experiment are performed , for example

– Toss a coin 6 times;Toss a coin 6 times;– Consider obtaining a head on a single toss as a success Consider obtaining a head on a single toss as a success

and obtaining a tail as a failure;and obtaining a tail as a failure;– Throw a die 10 times;Throw a die 10 times;– Consider obtaining a 6 on a single throw as a success, Consider obtaining a 6 on a single throw as a success,

and not obtaining a 6 as a failure.and not obtaining a 6 as a failure.

Example Example A coin a biased so that the probability of obtaining a A coin a biased so that the probability of obtaining a

head is . The coin is tossed four times. Find the probhead is . The coin is tossed four times. Find the probability of obtaining exactly two heads.ability of obtaining exactly two heads.

Example Example An ordinary die is thrown seven times. Find the probaAn ordinary die is thrown seven times. Find the proba

bility of obtaining exactly three sixes.bility of obtaining exactly three sixes.

Example Example The probability that a marksman hits a target is p and The probability that a marksman hits a target is p and

the probability that he misses is q, where q = 1 – p. the probability that he misses is q, where q = 1 – p. Write an expression for the probability that, in 10 shotWrite an expression for the probability that, in 10 shots, he hits the target 6 times.s, he hits the target 6 times.

If the probability that an experiment results in a successful outcome is p and the probability that the outcome is a failure is q, where q = 1 – p, and if X is the random variable ‘the number of successful outcomes in n independent trials’, then the probability function of X is given by

P(X = x) = for x = 0,1,2,…,n

Example Example If p is the probability of success and q = 1- p is the probability of faiIf p is the probability of success and q = 1- p is the probability of fai

lure, find the probability of 0,1,2,…,5 successes in 5 independent tlure, find the probability of 0,1,2,…,5 successes in 5 independent trials of the experiment. Comment your answer.rials of the experiment. Comment your answer.

In general:The values P(X =x) for x = 0,1,…,n can be obtained by considering the terms in the binomial expansion of (q + p)n, noting that q + p = 1(q + p)n =

1 = P(X = 0) + P(X =1) + P(X = 2) +…+ P(X= r) +….+P(X = n)

nnn

rrnnr

nnnnnn pqCpqCpqCpqCpqC 0222

111

00 ......

If X is distribution in this way, we writeIf X is distribution in this way, we write

– n and p are called the parameters of the distribun and p are called the parameters of the distribution.tion.

Sometimes, we will use b(x; n,p) to represenSometimes, we will use b(x; n,p) to represent the probability function when Xt the probability function when XBin(n,p). i.Bin(n,p). i.e. b(x; n,p) = P(X = x) = e. b(x; n,p) = P(X = x) =

X Bin(n,p) where n is the number of independent trials and p is the probability of a successful outcome in one trial

xxnnx pqC

Example Example – The probability that a person supports Party A is 0.6. Find the probThe probability that a person supports Party A is 0.6. Find the prob

ability that in a randomly selected sample of 8 voters there are ability that in a randomly selected sample of 8 voters there are (a) exactly 3 who support Party A, (a) exactly 3 who support Party A, (b) more than 5 who support Party A.(b) more than 5 who support Party A.

Example Example – A box contains a large number of red and yellow tulip bulbs in the rA box contains a large number of red and yellow tulip bulbs in the r

atio 1:3. Bulbs are picked at random from the box. How many bulbs atio 1:3. Bulbs are picked at random from the box. How many bulbs must be picked so that the probability that there is at least one red tmust be picked so that the probability that there is at least one red tulip bulb among them is greater than 0.95?ulip bulb among them is greater than 0.95?

6.2.2 6.2.2 Mean and Variance of the BinomiMean and Variance of the Binomial al DistributionDistribution–

– Proved it by yourself. :^) Proved it by yourself. :^)

= np2 = npq

Example Example – If the probability that it is find day is 0.4, find the expecIf the probability that it is find day is 0.4, find the expec

ted number of find days in a week, and the standard dted number of find days in a week, and the standard deviation.eviation.

Example Example – The random variable X is such that X The random variable X is such that X Bin(n,p) and E Bin(n,p) and E

(X) = 2, Var(X) = . Find the values of n and p, and P(X (X) = 2, Var(X) = . Find the values of n and p, and P(X = 2).= 2).

6.2.3 Application6.2.3 Application

C.W. C.W. Applications of Binomial distributionsApplications of Binomial distributionsThroughout this unit, daily lift examples and discussions are the essentThroughout this unit, daily lift examples and discussions are the essential features.ial features.

– Binomial DistributionBinomial DistributionQ1Q1The probability that a salesperson will sell a magzine subscription to soThe probability that a salesperson will sell a magzine subscription to so

meone who has been randomly selected from the telephone directory imeone who has been randomly selected from the telephone directory is 0.1. If the salesperson calls 6 individuals this evening, what is the pros 0.1. If the salesperson calls 6 individuals this evening, what is the probability thatbability that(I) There will be no subscriptions will be sold?(I) There will be no subscriptions will be sold?(II) Exactly 3 subscriptions will be sold?(II) Exactly 3 subscriptions will be sold?(III) At least 3 subscriptions will be sold?(III) At least 3 subscriptions will be sold?(IV) At most 3 subscriptions will be sold?(IV) At most 3 subscriptions will be sold?

6.3 THE GEOMETRIC DISTRIBUTI6.3 THE GEOMETRIC DISTRIBUTIONON

6.3.16.3.1 The Probability FunctionThe Probability Function

A geometric distribution arises when we have a sequence of indepenA geometric distribution arises when we have a sequence of independent trials, each with a definite probability p of success and probabilitdent trials, each with a definite probability p of success and probability q of failure, where q = 1 – p. Let X be the random variable ‘the numy q of failure, where q = 1 – p. Let X be the random variable ‘the number of trials up to and including the first success’.ber of trials up to and including the first success’.

Now, Now, P(X = 1) = P(success on the first trial) = pP(X = 1) = P(success on the first trial) = p P(X = 2) = P(failure on first trial, success on second) = q pP(X = 2) = P(failure on first trial, success on second) = q p P(X= 3) = ___________________________P(X= 3) = ___________________________ P(X = 4) = __________________________P(X = 4) = __________________________ …….. …….. P(X = x ) = __________________________P(X = x ) = __________________________

p is the parameter of the distribution.p is the parameter of the distribution.

If X is defined in this way, we writeIf X is defined in this way, we write

P(X = x) = qx – 1 p, x = 1,2,3,…… where q = 1 – p.

X Geo(p)

6.3.2 Mean and Variance6.3.2 Mean and Variance

Proved it by yourself. Proved it by yourself.

= and 2 = p

12p

q

Example Example – The probability that a marksman hits the bull’s eThe probability that a marksman hits the bull’s e

ye is 0.4 for each shot, and each shot is indepeye is 0.4 for each shot, and each shot is independent of all others. Findndent of all others. Find the probability that he hits the bull’s eye for the first tithe probability that he hits the bull’s eye for the first ti

me on his fourth attempt,me on his fourth attempt, the mean number of throws needed to hit the bull’s ethe mean number of throws needed to hit the bull’s e

ye, and the standard deviation,ye, and the standard deviation, the most common number of throws until he hits the the most common number of throws until he hits the

bull’s eye.bull’s eye.

ExampleExample– A coin is biased so that the probability of obtainiA coin is biased so that the probability of obtaini

ng a head is 0.6. If X is the random variable ‘the ng a head is 0.6. If X is the random variable ‘the number of tosses up to and including the first henumber of tosses up to and including the first head’, findad’, find P(X P(X 4), 4), P(X > 5),P(X > 5), The probability that more than 8 tosses will be requirThe probability that more than 8 tosses will be requir

ed to obtain a head, given the more than 5 tossed ared to obtain a head, given the more than 5 tossed are required.e required.

ExampleExample– In a particular board game a player can get out In a particular board game a player can get out

of jail only by obtaining two heads when she tosof jail only by obtaining two heads when she tosses two coins.ses two coins. Find the probability that more than 6 attempts are neFind the probability that more than 6 attempts are ne

eded to get out of jail.eded to get out of jail. What is the smallest value of n if there is to be at leasWhat is the smallest value of n if there is to be at leas

t a 90% chance of getting out of jail on or before the t a 90% chance of getting out of jail on or before the n th attempt.n th attempt.

C.W. Application of Geometric DistributionC.W. Application of Geometric Distribution

– 1)The probability that a student will pass a test on any tri1)The probability that a student will pass a test on any trial is 0.6. What is the probability that he will eventually pal is 0.6. What is the probability that he will eventually pass the test on the second trial?ass the test on the second trial?

– 2)Suppose the probability that Hong Kong Observatory 2)Suppose the probability that Hong Kong Observatory will make correct daily whether forecasts is 0.8. In the cwill make correct daily whether forecasts is 0.8. In the coming days, what is the probability that it will make the fioming days, what is the probability that it will make the first correct forecast on the fourth day?rst correct forecast on the fourth day?

6.4 THE POISSON DISTRIBUTION6.4 THE POISSON DISTRIBUTION

[see textbook][see textbook]

Example Example Verify that if XVerify that if XPo(Po(), then X is a random variable.), then X is a random variable.

Example Example If XIf XPo(Po() find (a) E(X), (b) E(X2), (c) Var(X).) find (a) E(X), (b) E(X2), (c) Var(X).

From above example , we can conclude that the MEAN anFrom above example , we can conclude that the MEAN and VARIANCE of the Poisson distribution are d VARIANCE of the Poisson distribution are and and respec respectively.tively.

C.W. Application of Poisson DisC.W. Application of Poisson Distributiontribution

– 1)The average number of claims per day mad1)The average number of claims per day made to the Insurance Company for damage or loe to the Insurance Company for damage or losses is 3.1. What is the probability that in any sses is 3.1. What is the probability that in any given daygiven day

fewer than 2 claims will be made?fewer than 2 claims will be made? exactly 2 claims will be made?exactly 2 claims will be made? 2 or more claims will be made?2 or more claims will be made? more than 2 claims will be made?more than 2 claims will be made?

– 2) Based on past experience, 1% of the teleph2) Based on past experience, 1% of the telephone bills mailed to house-holds in Hong Kong arone bills mailed to house-holds in Hong Kong are incorrect. If a sample of 10 bills is selected, fine incorrect. If a sample of 10 bills is selected, find the probability that at least one bill will be incod the probability that at least one bill will be incorrect. Do this using two probability distributions rrect. Do this using two probability distributions (the binomial and the Poisson) and briefly comp(the binomial and the Poisson) and briefly compare and explain your result.are and explain your result.