Taburan Poisson

8/11/2019 Taburan Poisson

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SPECI L PROB BILITY DISTRIBUTION II

JUHAIDAH BINTI JAMAL

FACULTY EDUCATION OF TECHNICAL

AND VOCATIONAL


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Poisson Distribution

The Poisson probability distribution describes the

number of times of a specified occurrence or event

that take place within a unit of time or space.

For instance, number of assignments submitted to a

lecturer in one semester, number of fatal accidents

per month in a manufacturing plant, number of phone

calls in one minute, number of rain day in a year.

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Ujikaji Poisson menghasilkan bilangan peristiwa kejayaanyang berlaku dalam suatu tempoh selang tertentu.

Selang tersebut adalah dalam bentuk seperti selang masa

yang diberi, rantau atau kawasan atau ruang tertentu.

Contoh pembolehubah rawak Poisson:

Bil. panggilan telefon yg tiba di sebuah pejabat dlm masa satu

minit

Bil. hujan lebat setahun,

Bil. kesalahan menaip di setiap muka surat

Bil. mikroorganisma dalam satu mlair kolam dan lain-lain.

Poisson Distribution


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Properties of Poisson Random

Variable

The experiment consists of counting the number X of times for a

particular event occurs during a given unit of time, or in a given

area or volume (or weight, distance, or any other unit of

measurement).

The probability that an event occurs in a given unit of time, area,

or volume is the same for all the units.

The number of event is independent.

The Poisson random variable is the number of success event.


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Example 3.10

Consider the number of patients arriving at the emergency ward of

a hospital during a one-hour interval. Is this example is Poisson

distribution?Solution:

1. The event is the arrival of a patient at the emergency ward,

the interval is one (an interval of time), and random.

2. The total number of patients who may arrive at this emergencyward during a one-hour interval may be 0,1,2,and it same for

the next one-hour interval.

3. The independence of events in this example means that

patients arrive individually and the arrival of any two (ormore) patients is not related.

4. The Poisson random variable is the number of patients arriving at

the emergency ward of a hospital during a one-hour interval.


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EXAMPLE (PG. 69)

a) P (X=1)

b) P (X2)

d) P (X2)

e) P (X4)

f) P (1X3)

g) P (1


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EXAMPLE 3.13 (PG. 71)

An average of three cars arrive at a highway tollgate

every minute. If this rate is approximated by a poisson

process, what is the probability that,

a) exactly five cars will arrive in a one minute period?

b) More than two cars will arrive in a one minute

period?

c) At least five cars will arrive in a two minute period?


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EXERCISE 1

Sebuah syarikat insurans mendapati bahawa min bilangan

tuntutan sebulan ialah 3. Apakah kebarangkalian bahawa bagi

sesuatu bulan syarikat itu mendapat 5 tuntutan? (0.1008)

EXERCISE 2Di sebuah perpustakaan, min bilangan org yg meminjam

buku dlm masa satu jam ialah 6. Apakah kebarangkalian

bahawa:

a) Tiada seorang pun meminjam buku dalam tempoh satuminit? (0.9048)

b) Sekurang-kurangnya seorang peminjam buku dalam tempoh

5 minit? (0.3935)


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How To Use Poisson

Probability Table

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Since Poisson distribution is

one of distribution for discrete

random variable, so we use thesame tips as in binomial

distribution


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TIPS TO USEBINOMIAL& POISSONPROBABILITY TABLE

Here are sometips how to

compute theprobability by

using table

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1) Pembolehubah rawak X mempunyai taburan Poisson

dengan min 12. Cari P (X 8)

2) X ~ P (6), find P(X10)

= 0.9105

= 0.0839


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Mean and variance of Poisson

Distribution


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The number of incorrectly dialed numbers

made per hour by a telephone sales

representative has Poisson distribution.

The probability she dials all the numbers

correctly is 0.82. find the meanof correctly

dialed numbers per hour.


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Poisson Approximation to Binomial

Distribution

(Taburan Poisson sebagai suatu penghampiran bagiTaburan Binomial)

If random variable is binomially distributed

with its parameter nandpthat fulfill thecondition n 30 and p < 0.1, then the

binomial distribution can be approximated

to Poisson distribution with = np


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Random variable X is binomially distributed

with n =100 and p= 0.02. Evaluate P(X1),

a) By using binomial distribution

b) By using Poisson approximation


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EXAMPLE

If there are 200 tpographical errors random

distributed in a 500-page manuscript, find theprobability that a given page contains exactly 3

errors.

Solution: First, find the mean number of errors. Since there 200

errors distributed over 500 pages, each page has an

average of

= 200 / 500 = 0.4 P(X=0.4)= 0.0072


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EXERCISE 1

Katakan kebarangkalian bahawa sejenis biji yang

tertentu tidak akan bercambah ialah 0.02. Jika

100biji ditanam, apakah kebarangkalian lebih

daripada 5 tidak akan bercambah.

a) dengan taburan Binomialb) dengan penghampiran Poisson


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EXERCISE 2

If approximately 2% of the people in a room of

200 people are left-handed, find the probability

that exactly 5 people there are left-handed.

i. Find mean (= n.p)ii. Find the probability that exactly 5 people there are

left-handed.


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EXERCISE 3

Di sebuah Jabatan, 0.5% daripada pekerjanya

menghampiri umur bersara. Cari kebarangkalian

bahawa lebih daripada 2 tetapi kurang daripada 5

orang pekerja menghampiri umur bersara sekiranya

jabatan ini mempunyai 1000 orang pekerja.

P(2


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EXERCISE 4

5% daripada bata yang diperbuat di sebuah kilang adalah

rapuh. Dengan penghampiran Poisson, cari kebarangkalian

bahawa suatu sampel 80 buah bata mengandungi 2 buah

bata rapuh.

Bata-bata yang dihasilkan dimasukkan ke dalam

kotak-kotak yang mengandungi 1000 buah bata setiapkotak. Bagi setiap kotak, 80 buah bata diperiksa secara

rawak dan kotak ini diluluskan sekiranya kurang daripada

dua buah bata didapati rapuh. Apakah kebarangkalian

bahawa sebuah kotak yang mengandungi 80 buah batarapuh diluluskan?

a) 0.1465

b) 0.0123


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EXERCISE 5

3% daripada anak burung gagak mati sebelum dapat

belajar terbang. Pada suatu tahun 500 ekor anak burung

gagak dilahirkan di sebuah kawasan. Dengan

menggunakan jadual, cari kebarangkalian bahawa:

a) Tidak lebih daripada sepuluh ekor anak burung gagak

mati sebelum dapat belajar terbang. (0.1185)

b) 12 ekor anak burung gagak mati sebelum dapat

belajar terbang. (0.0828)


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Thank you

Documents

Taburan Poisson