Download Chapter 3 - FAU Math

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2-8 Random Variables
Types of Random Variables
Chapter 3: Discrete Distribution
Chapter 4: Continuous Distribution
Exercises:
•
•
•
•
•
•
•
•
•
3.1: 3-3, 3-4, 3-7, 3-15
3.2: 3-17, 3-19, 3-23, 3-27, 3-33
3.3: 3-39, 3-41, 3-49, 3-51
3.4: 3-57, 3-59, 3-67
3.5: 3-77, 3-79, 3-83, 3-87,
3.6: 3-91, 3-93, 3-97, 3-103, 3-113
3.7: 3-119, 3-123, 3-127, 3-135
3.8: 3-141, 3-145, 3-151, 3-155
3.9: 3-175, 3-177, 3-185, 3-189, 3-201, 3-203, 3209
Discrete Probability Distribution
Camera 1
Camera 2
Camera 3
x
P(X=x)=f(x)
Fail
F
F
0
(.2)(.2)(.2)=0.008
Pass
F
F
1
(.8)(.2)(.2)=0.032
F
P
F
1
(.2)(.8)(.2)=0.032
F
F
P
1
0.032
P
P
F
2
(.8)(.8)(.2)=0.128
P
F
P
2
0.128
F
P
P
2
0.128
P
P
P
3
(.8)(.8)(.8)=.512
x
f(x)
0
0.008
1
0.096
2
0.384
3
0.512
3-1 Discrete Random Variables
Example 3-2 The time to recharge the flash is tested in 3 cell phone cameras.
P(a camera passes the test)=0.80 and cameras performs independently.
Let X=the number of cameras that pass the test.
What is the sample space of the test outcomes?
What are the possible values of X?
What is the probability distribution of X?
3-2 Probability Distributions and
Probability Mass Functions
Definition
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3-2 Probability Distributions and
Probability Mass Functions
3-2 Probability Distributions and
Probability Mass Functions
Example 3-4: There is a chance that a bit transmitted through a
digital transmission channel is received in error.
X=the number of bits in error in next
four bits transmitted
Figure 3-1 Probability distribution for bits in error.
Figure 3-2 Loadings at discrete points on a long, thin
beam.
Example 3-8
3-3 Cumulative Distribution Functions
Definition
Tree diagram
Recap
Possible value
p.m.f.
CDF
x
f(x)
F(x)
0
0.886
0.886
1
0.111
0.997
2
0.003
1.000
Total
1.000
3-4 Mean and Variance of a Discrete
Random Variable
Variance=measure of spread
Mean=Measure of centrality
Figure 3-5 A probability distribution can be viewed as a loading
with the mean equal to the balance point. Parts (a) and (b)
illustrate equal means, but Part (a) illustrates a larger variance.
f(x)
F(x) 1-1 corresponding
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3-4 Mean and Variance of a Discrete
Random Variable
Definition
3-4 Mean and Variance of a Discrete
Random Variable
Discrete Uniform distribution
Figure 3-6 The probability distribution illustrated in Parts (a)
and (b) differ even though they have equal means and equal
variances.
Equality of mean and variance does not imply equal distribution
Example 3-11
3-4 Expected Value of a Function
(h) of a Discrete Random Variable
First moment (mean): E(X)= x f(x)
Second moment: E(X2 )= x2 f (x)
3-5 Discrete Uniform Distribution
(skip)
Definition
Variance V(X)=E(X2)-[E(X)]2
3-5 Discrete Uniform Distribution
(skip)
Example 3-13
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3-5 Discrete Uniform Distribution
(skip)
3-5 Mean and Variance
Discrete Uniform Distribution
Figure 3-7 Probability mass function for a discrete uniform
random variable.
Example
Mean and variance for
• X: The number of 48 voice lines is uniformly
distributed over integers 0, …48
• Mean: µ=E(X)=(0+48)/2=24
• Variance: 2=Var(X)=[(48-0+1)2-1]/12=200
• Standard Deviation:  =(200)=14.14
• Bernoulli random variable
x f(x)
xf(x)
x2f(x)
0
1-p
0
0
1
p
p
p
E(X)=p
E(X2)=p
Variance=E(X2)-(E(X))2=p-p2=p(1-p)
Counting technique
• Permutation: The number of ways to order n
different objects is n!=n(n-1)*(n-2)*…*2*1
• Permutation of subset: The number of
permutations of subsets of r objects selected
from a set of n different objects is
Prn  n(n  1)(n  2)...(n  r  1) 
n!
(n  r )!
Examples
• 5!=5*4*3*2*1=120, 3!=3*2*1=6
• 8!=8*7*…*2*1=40320
8
8!
8! 40320
 3   3!(8  3)!  3!5!  6(120)  56
 
• Combination: The number of ways selecting
r objects from n objects is  n   C n  n!
r 
 
r
r!(n  r )!
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3-6 Binomial Distribution
Bernoulli trials and random variables: Examples
3-6 Binomial Distribution
Definition
3-6 Binomial Distribution
3-6 Binomial Distribution
(more examples, skip)
Random experiments and random variables
3-6 Binomial Distribution:
Mean and Variance
3-6 Example 1
Example 3-18
Figure 3-8 Binomial distributions for selected values of
n and p.
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3-6 Example 2: Use table II, p739-741
3-6 Binomial Distribution
(skip)
Example 3-19
Let X~Bin(5, .10) where, n=5, p=.10.
n x p=.10
Find
F(3)=P(X 3) =.9995
P(X=3)=P(X3)-P(X 2)=.9995-.9914=.0081
P(X3)=1-P(X<3)=1-P(X 2)=1-F(2)=1-.9914=.0086
P(1 <X<4)=P(1 <X  3)=P(X 3)-P(X 1)=.9995-.9185=.0810
Go to section 3-8 for Hypogeometric dist’n.
3-7.1 Geometric Distribution
(waiting for the first success)
Example 3-5 Geometric Distribution
Example 3-20
Example 3-5 (continued)
3-7.1 Geometric Distribution
Bernoulli trial: Random experiment which gives two and
only two basic outcomes
f(x)=
• Independent Bernoulli trials are conducted
• P(success)=p,
• X=the number of Bernoulli trials needed to obtain the first
success
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3-7.1 Geometric Distribution
3-7.1: Mean and Variance for Geometric Random Variable
Figure 3-9. Geometric distributions for selected values
of the parameter p.
Example: Coin spinning
3-7.1 Geometric Distribution
Example 3-21
Coin spinning example: Wait to observe first head.
X=#of spinning required. Then X~Geometric(1/2)
3-7 Geometric : Lack of memory property
3-7.2 Negative Binomial Distributions
Lack of Memory Property
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3-7.2 Negative Binomial Distributions
3-7.2 Negative Binomial Distributions
Figure 3-10.
Negative binomial
distributions for
selected values of
the parameters r
and p.
Figure 3-11. Negative binomial random variable
represented as a sum of geometric random variables.
3-7.2 Negative Binomial Distributions
3-7.2 Negative Binomial Distributions
Example 3-25
3-7.2 Negative Binomial Distributions
Example 3-25
3-8 Hypergeometric Distribution
Definition
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3-8 Hypergeometric Distribution
3-8 Hypergeometric Distribution
Example 3-27
Example 3-27
3-8 Hypergeometric Distribution
Definition
3-8 Hypergeometric Distribution
Finite Population Correction Factor
As n/N goes to zero, then (N-n)/(N-1) goes to one.
Note: Same mean and similar variance formulas to Binomial
Comparison of hypergeometric and binomial distributions.
3-9 Poisson Distribution
Hypogeometric: Sampling without replacement
Binomial: Sampling with replacement
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3.9 Mean and Variance
3-9 Poisson Distribution
Consistent Units
3-9 Example 3-33
3-9 Poisson Distribution
Example 3-33
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