discrete distribution 1

8/6/2019 Discrete Distribution 1

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DISCRETEDISCRETE

PROBABILITYPROBABILITYDISTRIBUTIONDISTRIBUTION


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Discrete Random Variables and

Probability Distributions

Mean, Variance and Expectation

Binomial Distribution

Poisson Distribution

CONTENT


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Arandom variable assigns a numerical value to eachoutcome in a sample space

2 types of random variable

Discrete the value is discrete, can be ordered andthere are gap between adjacent value

(Ex: number of something)

Continuous the possible values always contain an

interval (all the points between some two numbers)or its probability are given by areas under a curve.

(Ex: weight, age, salary, height, temperature)

DISCRETE RANDOM VARIABLES

& PROBABILITY DISTRIBUTIONS


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The probability distribution is a description thatgives the probability for each value of therandom variables.

It is often expressed in the format of a graph,table or formula.

The probability distribution of a discrete random

variableXis the probability density function (pdf)P(X) =P(X=x).

P(X) = P(X = x) = 1 , where the sum is over allthe possible values ofx.

THE DISCRETE

PROBABILITY DISTRIBUTION


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TABLE & GRAPH


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Event : toss a coin two times S = { HH, HT, TH, TT}

We can represent the sample space like this;

Outcome X = x P (X = x) Y = y P (Y = y)

where, X is a random variable which represents the number ofHin an outcome

Yis a random variable which represents the number ofTin an outcome

Thus, the probability distribution is given by:

X = x

P(X = x)

Example


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1. Represent graphically the probability distribution for the samplespace for tossing three coins.

2. 5 out of 20 bulbs in a box are defect. Chong choose 3 bulbs fromthe bulbs randomly. If X is a random variable which represents

the number of defect bulbs in an outcome, Find the probabilitydistribution for X and sketch the graph.

3. During the summer months, a rental agency keeps track of thenumber of chain saws it rents each day during a period of 90days. The number of saws rented per day is represented by thevariableX. The results are shown here. Compute the probability

P(X) for eachX, and construct a probability distribution and graphfor the data.

XNumber of days - the number of saws rented per day

0 45 1 30 2 15 Total 90

Example


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LetXbe a discrete random variable with pdfP(X) =P(X=x):

1. The mean (expected value) ofXis given by:E(X) =x = x P(X=x)

1. The variance is given by:

Var(X) = x = xP(X=x) x

2. The standard deviation is given by

Mean, Variance & Expectation


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1. A resistor in a certain circuit is specified to have aresistance in the range of 99-101 . An engineer obtain2 resistors. The probability that both of them meetspecification is 0.36, the probability that neither of them

meet specification is 0.16. LetXrepresents the number ofresistors that meet the specification. Find the probabilitydistribution, mean and standard deviation ofX.

2. Five balls numbered 0, 2, 4, 6 and 8 are placed in a bigbag. After the balls are mixed, one ball is selected, its

number is noted, and then it replaced. If this experimentis repeated many times, find the variance and standarddeviation of the number on the balls.

Examples


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Binomial Distribution

A Binomial distribution results from a procedurethat meets all the following requirements

The procedure has a fixed number of trials ( thesame trial is repeated)

The trials must be independent

Each trial must have outcomes classified into 2relevant categories only (success & failure)

The probability of success remains the same in alltrials

Example: toss a coin, Baby is born, True/false question, product, etc ...


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Binomial Experiment or not ?

An advertisement for Vantinclaims a 77% end oftreatment clinical success rate for flu sufferers. Vantinis given to 15 flu patients who are later checked to seeif the treatment was a success.

A study showed that 83% of the patients receiving livertransplants survived at least 3 years. The files of 6 liverrecipients were selected at random to see if eachpatients was still alive.

In a study of frequent fliers (those who made at least 3domestic trips or one foreign trip per year), it wasfound that 67% had an annual income over RM35000.12 frequent fliers are selected at random and theirincome level is determined.


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Notation for the Binomial Distribution

Then,Xhas the Binomial distribution with parameters n and p denoted byX~ Bin (n, p) which read as

Xis Binomial distributed with number of trials n and probability of success p


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Binomial Probability Formula


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Examples

A fair coin is tossed 10 times. LetXbe thenumber of heads that appear. What is thedistribution ofX?

A lot contains several thousandcomponents. 10 % of the components aredefective. 7 components are sampled fromthe lot. LetXrepresents the number ofdefective components in the sample. Whatis the distribution ofX?


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Solves problems involving linear inequalities

At least, minimum of, no less than

At most, maximum of, no more than

Is greater than, more than

Is less than, smaller than, fewer than

u

e

"


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Examples

Find the probability distribution of the randomvariableXifX~ Bin (10, 0.4).

Find also P(X= 5) and P(X< 2).Then find the mean and variance forX.

A fair die is rolled 8 times. Find the probability

that no more than 2 sixes comes up.Then findthe mean and variance forX.


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Examples

A survey found that, one out of five Malaysianssay he or she has visited a doctor in any givenmonth. If10 people are selected at random, find

the probability that exactly 3 will have visited adoctor last month.

A survey found that 30% of teenage consumers

receive their spending money from part timejobs. If 5 teenagers are selected at random, findthe probability that at least 3 of them will havepart time jobs.


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Solve Binomial problems by statistics table

Use Cumulative Binomials Probabilities Table

n number of trials p probability of success knumber of successes in n trials X It give P(X k) for various values ofn and p

Example: n = 2 , p = 0.3

Then P(X 1) = 0.9100

Then P(X= 1) = P(X 1) - P(X 0) = 0.9100 0.4900 = 0.4200

Then P(X 1) = 1 - P(X 1) = 1 - P(X 1) = 1- 0.9100 = 0.0900


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1. Given that n = 12 , p = 0.25. Then find

P(X 3)

P(X= 7)

P(X 5)

P(X< 2)

P(X> 10)

2. Given that n = 9 , p = 0.7. Then find

P(X 4) P(X= 8)

P(X 3)

P(X< 5)

P(X> 6)

Examples


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Example

A large industrial firm allows a discount on anyinvoice that is paid within 30 days. Of allinvoices, 10% receive the discount. In a companyaudit, 12 invoices are sampled at random.

What is probability that fewer than 4 of12 sampledinvoices receive the discount?

Then, what is probability that more than 1 of the12 sampled invoices received a discount.


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Example

A report shows that 5% ofAmericans are afraid being alonein a house at night. If a random sample of 20 Americans isselected, find the probability that

There are exactly 5 people in the sample who are afraidof being alone at night

There are at most 3 people in the sample who are afraidof being alone at night

There are at least 4 people in the sample who are afraidof being alone at night


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Poisson Distribution

The Poisson distribution is a discrete probabilitydistribution that applies to occurrences of some event overa specified interval ( time, volume, area etc..)

The random variableXis the number of occurrences of an

event over some interval

The occurrences must be random

The occurrences must be independent of each other

The occurrences must be uniformly distributed over theinterval being used

Example of Poisson distribution

1. The number of emergency call received by an ambulance control in an hour.

2. The number of vehicle approaching a bus stop in a 5 minutes interval.

3. The number of flaws in a meter length of material


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Poisson Probability Formula

, mean number of occurrences in the given interval is knownand finite

Then the variableXis said to be

Poisson distributed with mean

X~ Po ()


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Example

A student finds that the average number of amoebasin 10 ml of ponds water from a particular pond is 4.Assuming that the number of amoebas follows a

Poisson distribution, find the probability that in a 10ml sample,

there are exactly 5 amoebas

there are no amoebas

there are fewer than three amoebas


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On average, the school photocopier breaks down 8times during the school week (Monday - Friday).Assume that the number of breakdowns can be

modeled by a Poisson distribution.Find the probability that it breakdowns,

5 times in a given week

Once on Monday

8 times in a fortnight

Example


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Solve Poisson problems by statistics table

Given thatX~ Po (1.6). Use cumulative Poissonprobabilities table to find

P(X 6)

P(X= 5) P(X 3)

P(X< 1)

P(X> 10)

Find also the smallest integer n such thatP(X> n) < 0.01


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A sales firm receives, on the average, three calls perhour on its toll-free number. For any given hour, findthe probability that it will receive the following:

At most three calls

At least three calls

5 or more calls

Example


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The number of accidents occurring in a weak in acertain factory follows a Poisson distribution withvariance 3.2.

Find the probability that in a given fortnight,

exactly seven accidents happen.

More than 5 accidents happen.

Example


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Using the Poisson distribution as an

approximation to the Binomial

distribution

When n is large (n > 50) and p is small (p < 0.1),the Binomial distributionX~ Bin (n, p) can beapproximated using a Poisson distribution withX~ Po () where mean, = np < 5.

The larger the value ofn and the smaller the valueofp, the better the approximation.


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Eggs are packed into boxes of 500. On average 0.7 %of the eggs are found to be broken when the eggs areunpacked.

Find the probability that in a box of 500 eggs,

Exactly three are broken

At least two are broken

Example


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If 2% of the people in a room of 200 people are left-handed, find the probability that

exactly five people are left-handed.

At least two people are left-handed.

At most seven people are left-handed.

Example


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A probability distribution can be graphed, and the mean,variance and standard deviation can be found.

Binomial and Poisson distribution are two commonly usedprobability distributions.

Binomial distribution is used when there are only 2independent outcomes (success or failure) for anexperiment, there are fixed number of trials and the

probability is the same for each trial.

Poisson distribution is used when an independent eventoccur over a period of time, area or volume.

SUMMARY

discrete distribution 1

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