Download random variable.

1
2
3
3-1 Introduction
• Experiment  measurement
• Random component  the measurement might differ
in day-to-day replicates because of small variations in
variables that are not controlled in our experiment
• Random experiment  an experiment that can result
in different outcomes, even though it is repeated in the
same manner every time.
4
3-1 Introduction
• No matter how carefully our experiment is designed,
variations often occur.
• Our goal is to understand, quantify, and model the type
of variations that we often encounter.
•When we incorporate the variation into our thinking
and analyses, we can make informed judgments from
our results that are not invalidated by the variation.
5
3-1 Introduction
Physical universe
Newton’s laws
6
3-1 Introduction
• We discuss models that allow for variations in the
outputs of a system, even though the variables that we
control are not purposely changed during our study.
• A conceptual model that incorporates uncontrollable
variables (noise) that combine with the controllable
variables to produce the output of our system.
• Because of the noise, the same settings for the
controllable variables do not result in identical outputs
7
every time the system is measured.
3-1 Introduction
8
3-1 Introduction
• Example: Measuring current in a copper wire
• Ohm’s law: Current = Voltage/resistance
• A suitable approximation
9
3-1 Introduction
10
3-2 Random Variables
• In an experiment, a measurement is usually
denoted by a variable such as X.
• In a random experiment, a variable whose
measured value can change (from one replicate of
the experiment to another) is referred to as a
random variable.
11
3-2 Random Variables
12
3-3 Probability
• Used to quantify likelihood or chance
• Used to represent risk or uncertainty in engineering
applications
•Can be interpreted as our degree of belief or
relative frequency
13
3-3 Probability
• Probability statements describe the likelihood that
particular values occur.
• The likelihood is quantified by assigning a number
from the interval [0, 1] to the set of values (or a
percentage from 0 to 100%).
• Higher numbers indicate that the set of values is
more likely.
14
3-3 Probability
• A probability is usually expressed in terms of a
random variable.
• For the part length example, X denotes the part
length and the probability statement can be written
in either of the following forms
• Both equations state that the probability that the
random variable X assumes a value in [10.8, 11.2] is
0.25.
15
3-3 Probability
Complement of an Event
• Given a set E, the complement of E is the set of
elements that are not in E. The complement is
denoted as E’.
Mutually Exclusive Events
• The sets E1 , E2 ,...,Ek are mutually exclusive if
the
intersection of any pair is empty. That is, each
element is in one and only one of the sets E1 , E2
16
,...,Ek .
3-3 Probability
Probability Properties
17
3-3 Probability
Events
• A measured value is not always obtained from an
experiment. Sometimes, the result is only classified
(into one of several possible categories).
• These categories are often referred to as events.
Illustrations
•The current measurement might only be
recorded as low, medium, or high; a manufactured
electronic component might be classified only as
defective or not; and either a message is sent through a
18
network or not.
3-4 Continuous Random Variables
3-4.1 Probability Density Function
• The probability distribution or simply distribution
of a random variable X is a description of the set of
the probabilities associated with the possible values
for X.
19
3-4 Continuous Random Variables
3-4.1 Probability Density Function
20
3-4 Continuous Random Variables
3-4.1 Probability Density Function
21
3-4 Continuous Random Variables
3-4.1 Probability Density Function
22
3-4 Continuous Random Variables
3-4.1 Probability Density Function
23
3-4 Continuous Random Variables
24
3-4 Continuous Random Variables
25
3-4 Continuous Random Variables
3-4.2 Cumulative Distribution Function
26
3-4 Continuous Random Variables
27
3-4 Continuous Random Variables
28
3-4 Continuous Random Variables
29
3-4 Continuous Random Variables
3-4.3 Mean and Variance
30
3-4 Continuous Random Variables
31
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
Undoubtedly, the most widely used model for the
distribution of a random variable is a normal
distribution.
Whenever a random experiment is replicated, the
random variable that equals the average result over
the replicates tends to have a normal distribution as
the number of replicates becomes large.
32
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
That is,
Given random variables X1,X2,…,Xk, Y = (1/n)iXi
tends to have a normal distribution as k.
• Central limit theorem
• Gaussian distribution
33
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
34
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
35
3-5 Important Continuous Distributions
36
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
37
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
38
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
39
3-5 Important Continuous Distributions
P(Z>1.26)
P(Z<-4.6)
P(Z<-0.86)
Find z s.t. P(Z>z)=0.05
P(Z>-1.37)
Find z s.t. P(-z<Z<z)=0.99
P(-1.25<Z<0.37)
40
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
41
3-5 Important Continuous Distributions
3-5.1 Normal Distribution
42
3-5 Important Continuous Distributions
43
3-5 Important Continuous Distributions
44
3-5 Important Continuous Distributions
3-5.2 Lognormal Distribution
45
3-5 Important Continuous Distributions
3-5.2 Lognormal Distribution
46
3-5 Important Continuous Distributions
3-5.3 Gamma Distribution
47
3-5 Important Continuous Distributions
3-5.3 Gamma Distribution
48
3-5 Important Continuous Distributions
3-5.3 Gamma Distribution
49
3-5 Important Continuous Distributions
3-5.4 Weibull Distribution
50
3-5 Important Continuous Distributions
3-5.4 Weibull Distribution
51
3-5 Important Continuous Distributions
3-5.4 Weibull Distribution
52
3-6 Probability Plots
3-6.1 Normal Probability Plots
• How do we know if a normal distribution is a reasonable
model for data?
• Probability plotting is a graphical method for determining
whether sample data conform to a hypothesized
distribution based on a subjective visual examination of the
data.
• Probability plotting typically uses special graph paper, known
as probability paper, that has been designed for the
hypothesized distribution. Probability paper is widely
available for the normal, lognormal, Weibull, and various chisquare and gamma distributions.
53
3-6 Probability Plots
Construction
• Sample data: x1,x2,..,xn.
• Ranked from smallest to largest: x(1),x(2),..,x(n).
• Suppose that the cdf of the normal probability paper
is F(x).
• Let F(x(j))=(j-0.5)/n.
•If the resulting curve is close to a straight line, then
we say that the data has the hypothesized probability
model.
54
3-6 Probability Plots
3-6.1 Normal Probability Plots
55
3-6 Probability Plots
An ordinary graph probability
• Plot the standarized normal scores zj against x(j)
• Z: standardized normal distribution
• z1,..,zn: P(Z  zj) = (j-0.5)/n.
56
3-6 Probability Plots
3-6.1 Normal Probability Plots
57
3-6 Probability Plots
3-6.2 Other Probability Plots
58
3-6 Probability Plots
3-6.2 Other Probability Plots
59
3-6 Probability Plots
3-6.2 Other Probability Plots
60
3-6 Probability Plots
3-6.2 Other Probability Plots
61
3-7 Discrete Random Variables
• Only measurements at discrete points are
possible
62
3-7 Discrete Random Variables
• Example 3-18 (Finite case)
• There is a chance that a bit transmitted through a
digital transmission channel is received in error. Let X
equal the number of bits in error in the next 4 bits
transmitted. The possible value for X are {0,1,2,3,4}.
Based on a model for the errors that is presented in the
following section, probabilities for these values will be
determined.
63
3-7 Discrete Random Variables
3-7.1 Probability Mass Function
64
3-7 Discrete Random Variables
3-7.1 Probability Mass Function (Infinite case)
65
3-7 Discrete Random Variables
3-7.1 Probability Mass Function
66
3-7 Discrete Random Variables
3-7.2 Cumulative Distribution Function
67
3-7 Discrete Random Variables
3-7.2 Cumulative Distribution Function
18
68
3-7 Discrete Random Variables
3-7.3 Mean and Variance
69
3-7 Discrete Random Variables
3-7.3 Mean and Variance
70
3-7 Discrete Random Variables
3-7.3 Mean and Variance
71
3-8 Binomial Distribution
• Consider the following random experiments and random
variables.
• Flip a fair coin 10 times. X = # of heads obtained.
• A worn machine tool produces 1% defective parts. X = #
of defective parts in the next 25 parts produced.
• Water quality samples contain high levels of organic solids
in 10% of the tests. X = # of samples high in organic solids
in the next 18 tested.
72
3-8 Binomial Distribution
• A trial with only two possible outcomes is used so
frequently as a building block of a random experiment
that it is called a Bernoulli trial.
• It is usually assumed that the trials that constitute the
random experiment are independent. This implies that
the outcome from one trial has no effect on the
outcome to be obtained from any other trial.
• Furthermore, it is often reasonable to assume that the
probability of a success on each trial is constant.
73
3-8 Binomial Distribution
• Consider the following random experiments and
random variables.
•
•
Flip a coin 10 times. Let X = the number of heads obtained.
Of all bits transmitted through a digital transmission
channel, 10% are received in error. Let X = the number of
bits in error in the next 4 bits transmitted.
Do they meet the following criteria:
1. Does the experiment consist of Bernoulli
trials?
2. Are the trials that constitute the random
experiment are independent?
3. Is probability of a success on each trial is
constant?
74
3-8 Binomial Distribution
75
3-8 Binomial Distribution
76
3-8 Binomial Distribution
77
3-9 Poisson Process
• Consider e-mail messages that arrive at a mail server
on a computer network.
• An example of events (message arriving) that occur
randomly in an interval (time).
• The number of events over an interval (e.g. # of messages
that arrive in 1 hour) is a discrete random variable that is
often modeled by a Poisson distribution.
• The length of the interval between events (time between
two messages) is often modeled by an exponential
distribution.
78
3-9 Poisson Process
79
3-9 Poisson Process
3-9.1 Poisson Distribution
80
3-9 Poisson Process
Example 3-27
• Flaws occur at random along the length of a thin copper
wire.
• Let X denote the random variable that counts the number
of flaws in a length of L mm of wire and suppose that the
average number of flaws in L mm is .
• Partition the length of wire into n subintervals of small
length.
• If the subinterval chosen is small enough, the probability
that more than one flaw occurs in the subinterval is
negligible.
81
3-9 Poisson Process
Example 3-27
• We interpret that flaws occur at random to imply that every
subinterval has the same probability of containing a flaw –
say p.
• Assume that the probability that a subinterval contains a
flaw is independent of other subintervals.
• Model the distribution X as approximately a binomial
random variable.
• E(X)==np  p= /n.
• With small enough subintervals, n is very large and p is
very small. (similar to Example 3-26).
82
3-9 Poisson Process
3-9.1 Poisson Distribution
83
3-9 Poisson Process
3-9.1 Poisson Distribution
84
3-9 Poisson Process
3-9.1 Poisson Distribution
85
3-9 Poisson Process
3-9.1 Poisson Distribution
86
3-9 Poisson Process
3-9.2 Exponential Distribution
• The discussion of the Poisson distribution defined a random
variable to be the number of flaws along a length of copper wire.
The distance between flaws is another random variable that is
often of interest.
• Let the random variable X denote the length from any starting
point on the wire until a flaw is detected.
87
3-9 Poisson Process
3-9.2 Exponential Distribution
• X: the length from any starting point on the wire until a flaw is
detected.
•
•As you might expect, the distribution of X can be obtained from
knowledge of the distribution of the number of flaws. The key to
the relationship is the following concept:
The distance to the first flaw exceeds h millimeters if and only
if there are no flaws within a length of h millimeters—simple,
but sufficient for an analysis of the distribution of X.
88
3-9 Poisson Process
3-9.2 Exponential Distribution
• Assume that the average number of flaws is  per mm.
• N: number of flaws in x mm of wire.
• N is a Poisson distribution with mean x.
e  x (  x ) 0
 e  x
• Pr(X>x) = P(N=0) =
0!
• So F(x) = Pr(Xx) =1-e-x.
• Pdf f(x)= e-
89
3-9 Poisson Process
3-9.2 Exponential Distribution
90
3-9 Poisson Process
3-9.2 Exponential Distribution
91
3-9 Poisson Process
3-9.2 Exponential Distribution
Example 3-30:
• In a large corporate computer net work, user log-ons to the
system can be modeled as a Poisson process with a mean
of 25 log-ons per hour.
• What is the probability that there are no log-ons in an
interval of 6 minutes?
• Solution:
• X: the time in hours from the start of the interval until the
first log-on.
• X: an exponential distribution with =25 log-ons per hour.
92
3-9 Poisson Process
3-9.2 Exponential Distribution
Example 3-30:
• 6 min = 0.1 hour.

 25 x
 2.5
P
(
X

0
.
1
)

25
e
dx

e
 0.082
•

0.1
93
3-9 Poisson Process
3-9.2 Exponential Distribution
94
3-9 Poisson Process
3-9.2 Exponential Distribution
• The exponential distribution is often used in reliability studies as
the model for the time until failure of a device.
• For example, the lifetime of a semiconductor chip might be
modeled as an exponential random variable with a mean of 40,000
hours.
• The lack of memory property of the exponential distribution
implies that the device does not wear out.
•The lifetime of a device with failures caused by random shocks
might be appropriately modeled as an exponential random variable.
95
3-9 Poisson Process
3-9.2 Exponential Distribution
• However, the lifetime of a device that suffers slow mechanical
wear, such as bearing wear, is better modeled by a distribution that
does not lack memory.
96
3-10 Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Binomial
97
3-10 Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Binomial
98
3-10 Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Binomial
99
3-10 Normal Approximation to the Binomial
and Poisson Distributions
Normal Approximation to the Poisson
• Poisson distribution is developed as the limit of a binomial
distribution as the number of trials increased to infinity.
• The normal distribution can also be used to approximate
probabilities of a Poisson random variable.
100
3-11 More Than One Random Variable
and Independence
3-11.1 Joint Distributions
101
3-11 More Than One Random Variable
and Independence
3-11.1 Joint Distributions
102
3-11 More Than One Random Variable
and Independence
3-11.1 Joint Distributions
103
3-11 More Than One Random Variable
and Independence
3-11.1 Joint Distributions
104
3-11 More Than One Random Variable
and Independence
3-11.2 Independence
105
3-11 More Than One Random Variable
and Independence
3-11.2 Independence
106
3-11 More Than One Random Variable
and Independence
3-11.2 Independence
107
3-11 More Than One Random Variable
and Independence
3-11.2 Independence
108
3-12 Functions of Random Variables
109
3-12 Functions of Random Variables
3-12.1 Linear Combinations of Independent
Random Variables
110
3-12 Functions of Random Variables
3-12.1 Linear Combinations of Independent
Normal Random Variables
111
3-12 Functions of Random Variables
3-12.1 Linear Combinations of Independent
Normal Random Variables
112
3-12 Functions of Random Variables
3-12.2 What If the Random Variables Are Not
Independent?
Recall that the inner product!
113
3-12 Functions of Random Variables
3-12.2 What If the Random Variables Are Not
Independent?
114
3-12 Functions of Random Variables
3-12.3 What If the Function Is Nonlinear?
115
3-12 Functions of Random Variables
3-12.3 What If the Function Is Nonlinear?
116
3-12 Functions of Random Variables
3-12.3 What If the Function Is Nonlinear?
117
3-13 Random Samples, Statistics, and
The Central Limit Theorem
118
3-13 Random Samples, Statistics, and
The Central Limit Theorem
Central Limit Theorem
119
3-13 Random Samples, Statistics, and
The Central Limit Theorem
120
3-13 Random Samples, Statistics, and
The Central Limit Theorem
121
3-13 Random Samples, Statistics, and
The Central Limit Theorem
122
123