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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.
2
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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
12
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
13
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14
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15
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16
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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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EXAMPLE 3.17 (PG. 73)
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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EXAMPLE 3.18 (PG. 74)
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