(4) Some Discrete Probability Distributions
Transcript of (4) Some Discrete Probability Distributions
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324 Stat
Lecture Notes
(4) Some Discrete
Probability
Distributions( Book: Chapter 5 ,pg 143)
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4.1 Discrete Uniform
Distribution:
If the random variable X assume the
values with equal probabilities, then the
discrete uniform distribution is given by:
)1(0
,...,,,1
),( 21
elsewhere
xxxXK
KXP K
Discrete Uniform is
not in the book, it
should be studied
from the notes
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Theorem:
The mean and variance of the discrete uniform
distribution are:),( KXP
)2(
)(
, 1
2
21
K
X
K
XK
i
K
i
i
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EX (1):
When a die is tossed once, each element of the sample space occurs with probability 1/6. Therefore we have a uniform distribution with:
Find:
1.
2.
3.
}6,5,4,3,2,1{S
6,5,4,3,2,1,6
1)6,( XXP
( 3)P x
(1 4)P x
(3 6)P x
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Solution:
1.
2.
3.
(1 4)P x
1 1 1 3( 1) ( 2) ( 3)
6 6 6 6P x P x P x
2( 3) ( 1) ( 2)
6P x P x P x
2(3 6) ( 4) ( 5)
6P x P x P x
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EX (2):
Solution:
For example (1): Find 2 and
1
2
2 1
2 2 2 2 2 2
1 2 3 4 5 63.5
6
( )
(1 3.5) (2 3.5) (3 3.5) (4 3.5) (5 3.5) (6 3.5) 35
6 12
k
i
i
k
i
i
X
k
X
k
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The Bernoulli Process:
Bernoulli trials are an experiment with:
1) Only two possible outcomes.
2) Labeled as success (S), failure (F).
3) Probability of success , probability of failure .
p1 ( 1)q p p q
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Binomial Distribution:
* The Binomial trials must possess the following properties:
1) The experiment consists of n repeated trials.
2) Each trial results in an outcome that may be classified as a success or a failure.
3) The probability of success denoted by premains constant from trial to trial.
4) The repeated trials are independent.
5) The parameters of binomial are n,p.
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Binomial Distribution:
The probability distribution of the binomial random variable X, the number of successes in n independent trials is:
( , , ) , 0,1,2,...., (5)X n Xn
b x n p p q x nx
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Theorem:
The mean and the variance of the
binomial distribution b(x, n, p) are:2 (6)np and npq
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EX (4): According to a survey by the Administrative
Management society, 1/3 of U.S. companies give employees four weeks of vacation after they have been with the company for 15 years. Find the probability that among 6 companies surveyed at random, the number that gives employees 4 weeks of vacation after 15 years of employment is:
a) anywhere from 2 to 5;
b) fewer than 3;
c) at most 1;
d) at least 5;
e) greater than 2;
f) calculate 2and
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Solution:
2 4 3 3 4 2 5 1
6, 1/3, 2 /3
( ) (2 5) ( 2) ( 3) ( 4) ( 5)
6 6 6 61 2 1 2 1 2 1 2
3 3 3 3 3 3 3 32 3 4 5
240 160 60 12 472
729 729 729 729
n p q
a P x P x P x P x P x
0.647729
0 6 1 5 2 4
( ) ( 3) ( 0) ( 1) ( 2)
6 6 61 2 1 2 1 2
3 3 3 3 3 30 1 2
64 192 240 4960.68
729 729 729 729
b P x P x P x P x
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0 6 1 5
( ) ( 1) ( 0) ( 1)
6 61 2 1 2
3 3 3 30 1
64 192 2560.351
729 729 729
c P x P x p x
5 1 6 0
( ) ( 5) ( 5) ( 6)
6 61 2 1 2
3 3 3 35 6
12 1 130.018
729 729 729
d P x P x P x
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0 6 1 5 2 4
( ) ( 2) 1 ( 2) ( 0) ( 1) ( 2)
6 6 61 2 1 2 1 21
3 3 3 3 3 30 1 2
64 192 240 496 2331 1 0.319
729 729 729 729 729
e P x P x P x P x P x
2
1 6( ) (6)( ) 2
3 3
1 2 12(6)( )( ) 1.333
3 3 9
f np
npq
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EX (5):
The probability that a person suffering from headache will obtain relief with a particular drug is 0.9. Three randomly selected sufferers from headache are given the drug. Find the probability that the number obtaining relief will be:
a) exactly zero;
b) at most one;
c) more than one;
d) two or fewer;
e) Calculate 2and
See Ex 5.2 pg 146
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Solution:
3, 0.9, 0.1n p q
0 3
3( ) ( 0) 0.9 0.1 0.001
0a P X
0 3 1 2
( ) ( 1) ( 0) ( 1)
3 30.9 0.1 0.9 0.1
0 1
0.001 0.027 0.028
b P x P x P x
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0 3 1 2
( ) ( 1) 1 ( 1) 1 [ ( 0) ( 1)]
3 31 [ 0.9 0.1 0.9 0.1
0 1
1 (0.001 0.027) 1 0.028 0.972
c P X p x P x P x
0 3 1 2 2 1
( ) ( 2) ( 0) ( 1) ( 2)
3 3 30.9 0.1 0.9 0.1 0.9 0.1
0 1 2
0.001 0.027 0.243 0.271
d P x P x P x P x