Download Chapter 5: Discrete Distributions

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Chapter 5: Discrete Distributions
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Section 5.2 Binomial Distribution
A binomial experiment is one yielding exactly one of two possible outcomes– “success” and
“failure”.
Properties of a binomial experiment
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There are n independent trials; each one results in either a success (S) or failure (F).
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The probability of a success, p, remains constant over trials.
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The random variable of interest is X, the number of successes in n trials.
Definition: A random variable X is said to have a binomial distribution if
n
P(X = x) =   p x q n − x , x = 0, 1, 2, …, n,
 x
where p = probability of success and q = 1 – p = probability of failure.
The mean, variance, and standard deviation of a binomial random variable X are
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Mean = E(X) = µ = np
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Variance = σ 2 = npq
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Standard deviation = σ = npq
An example of binomial experiment is selection with replacement, Flipping coins, throwing dice.
Example: Ex 5.22 page 192
Use the binomial Tables starting from page 569. For n = 5 and p = 0.8 go to page 570.
P(X = 5) = 1 – P(X ≤ 4) = 1 – 0.672 = 0.328
P(X = 4) = P(X ≤ 4) – P(X ≤ 3) = 0.672 – 0.263 = 0.409
P(X < 2) = P(X ≤ 1) = 0.007
k
P(X ≤ k) [p = 0.8]
0
0.000
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1
0.007
2
0.058
3
0.263
4
0.672
5
1.000
Example: Ex 5.12 page 190
Supplementary #9
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Section 5.4 The Hypergeometric Distribution
This is similar to the binomial distribution except that the trials are no more independent.
Definition: A random variable X is said to follow the hypergeometric distribution if
 M  N − M 
 

k
n−k 
P(X = k) =  
N
 
n
for values of k depending on N, M, and n.
The mean and variance of the hypergeometric distribution are
µ = np, where p =
M
N −n
 N −n
and =
is called the finite σ 2 np(1 − p) 
 . The quantity
N
N −1
 N −1 
population correction factor.
Selection without replacement is an example of the hypergeometric distribution. Here, N is finite
but in the case of binomial distribution, N is infinite.
Example:
A box contains 4 white balls and 5 black balls.
a) If you draw 3 balls together, what is the probability that two is white and one is black?
b) If X is a random variable denotes to the number of white balls drawn. Write the
probability distribution table for this random variable.
c) Find the expected value, the variance, and the standard deviation of X.
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Ex 5.44 page 202
Ex 5.46 page 202
Find the mean and variance of X in Ex 5.46.
Supplementary #10