Download CHAPTER 4—EXPECTED VALUE FUNCTION

Chapter 4--Expected Value Function.Doc
STATISTICS 301—APPLIED STATISTICS, Statistics for Engineers and Scientists, Walpole, Myers, Myers, and Ye, Prentice Hall
GOAL
In this section we introduce the EXPECTED VALUE FUNCTION which can be used to
summarize random variables. The expected value function is something that is seen over and
over in more advanced statistics courses.
REVIEW OF FUNCTIONS
Consider f(x) = 5x2.
What is f(2)?
f( ¼ )?
f( 3 )?
f( ½ y )?
FACT: All functions are of the form:
f( argument) = operation on the argument
That is: f(  ) = 5 (  )2.
Motivation
#1: Why do we need to summarize RV’s?
#2: What do we summarize about RV’s?
Consider the following example.
Suppose I need to talk to someone and all I have is my cell phone, but I’ve used all of my
allotted minutes. However, I have Verizon and I have unlimited “in-network” calling on my
plan. Suppose I start asking people to determine if they are also a Verizon customer.
Let X = the number of cell phone customers asked until I get the first Verizon
customer.
Note that X is discrete since SX = { 1, 2, 3, …. } and X is one of those countably infinite
discrete RV’s.
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Let’s assume that Verizon has a 50% share of the cell phone market, so that I have a 50%
chance of getting a Verizon customer each time I ask a person their cell phone company.
Below is the probability function for X, f(x) = 0.5x (Why?), in tabular and graphical format.
x
f(x)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
…
0.5
0.25
0.125
0.0625
0.03125
0.015625
0.007813
0.003906
0.001953
0.000977
0.000488
0.000244
0.000122
6.1E-05
3.05E-05
1.53E-05
7.63E-06
3.81E-06
1.91E-06
9.54E-07
4.77E-07
2.38E-07
1.19E-07
5.96E-08
2.98E-08
1.49E-08
7.45E-09
3.73E-09
1.86E-09
9.31E-10
…
While the above probability function for X gives us complete information about the RV, it’s
almost TOO much. It would be better if we could summarize the important features of the
distribution of our RV, rather than having to wade through all of the values and probabilities
of X.
Since these numbers form a distribution and we know that we can summarize the
distribution using numbers that represent the Center, Spread, and Shape. Also recall that
the set of values a RV can take on represents a population. Hence when we summarize this
population distribution of values for the RV, we will use
The MEAN to represent the CENTER
The STANDARD DEVIATION to represent the SPREAD
The KIND OF RV will determine it’s SHAPE!
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The following example will illustrate why, when we are summarizing a distribution, we need to
summarize, at least, three different characteristics; just one won’t do it!!!
0.20
Cauchy
0.0
0.05
0.10
0.15
0.2
0.1
Normal
0.3
0.25
0.30
0.4
Below are examples of four VERY different continuous distributions, as denoted by their
pdf’s.
-3
-2
-1
0
1
2
3
-3
-2
-1
x
0
1
2
3
0.15
Uniform
0.10
15
0
0.0
5
0.05
10
Gamma
20
25
0.20
30
0.25
x
-3
-2
-1
0
x
1
2
3
-3
-2
-1
0
1
2
3
x
Each of the four distributions have centers that are pretty clearly zero, but their SPREADs
and SHAPE differ markedly! Hence we need to summarize ALL three aspects of
distribution.
To this end we will define the Expected Value function. This function of the values of the
RV as well as the probability function or pdf will allow us to obtain summary measures of the
center and spread of a RV’s distribution.
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Defn: Let X be any RV with probability distribution, fX(x). The Expected Value Function is
defined to be
E [ g(X) ] =

  g(x) [fX (x)], if X is discrete
 all x values


  g(x) [fX (x)dx], if X is continuous
-
Notes and Comments
1. E [] is a function, whose argument is .
2. The E [] function is interpreted as an “averaging” function. That is, whatever you
find the expected value of, the resulting expected value is the “average” of that
value.
So whatever E[X] equals would be interpreted as the “average” or “mean” value of the
possible values of X.
2. Defn: A simple weighted average is
 weights*values   weights .
Hence the Expectation Function is an “averaging” function, but unlike the usual
averaging
xn
the Expectation Function is a “weighted averaging” function. Why?
The weights are nothing more than the probability distribution values, the fX(x).
Furthermore, when we calculate an expected of any function of the RV, we obtain an
“average” value of that function of the RV. So if we were to calculate the expected
value of X, E [ X ], we would obtain an “average” value of the random variable. Sound
familiar?
3. The expected value of various functions of a RV will provide summary features of the
distribution of the RV.
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EXAMPLE #1
Let X = # of good components when 3 components are selected randomly from a box
containing 7 components, 3 of which are bad. X is discrete with probability distribution
given by
4!
3!
 4  3 
 x  3-x 
 = X!(4-X)! (3-X)!(3-(3-X))! , for x=0, 1, 2, 3
fX (x) =  
7!
7
3
3!4!
 
In tabular form we have
x
fX(x)
0
1
2
3
Find the expected value of X.
Find the expected value of X .
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EXAMPLE #2
Devore/Farnum—X = amount of gravel (100tons) sold per week by supply company.
fX (x) = 1.5(1-x2 ), 0 < x < 1
If the company makes $2.50 profit on each ton sold, what is the company’s
expected/average/mean profit per week?
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USES OF EXPECTED VALUES
1. Distributional Center—MEAN
Defn: The MEAN of a RV, X, is denoted by X and defined as X = E [ X ].
Interpretation: The use of the word MEAN is very important since we are dealing
with the entire population of values of the RV and hence when we calculate the
measure of center similar to the average we refer to it as the MEAN!
Recall that the expected value function is an averaging function, so that the E [ X ]
is the “average” value of X, but not in the traditional “average” sense.
2. Distributional Spread—VARIANCE
Defn: The VARIANCE of a RV, X, is denoted by 2X and defined as
2X = E [ ( X - X )2 ].
While the above is a definitional form of the variance of the RV, rarely the variance
calculated in this manner. Rather we use the computational form
2X = E [ X2 ] – ( X )2 .
Interpretation: The variance of a RV is the “average” squared distance of values of
the RV to the mean.
Defn: The STANDARD DEVIATION of a RV, X, is denoted by X and is simply the
positive square root of the variance X = √2X .
Note: The standard deviation is used more typically since the units associated with
SD are the same as the RV, whereas the variance is measured in squared units of
the RV.
3. Other Distributional Measures
While they can be defined, we will not discuss the other numerical measures of the
distribution, for example, measures of symmetry since these characteristics are
typically known as part of the “type” or “kind” of RV we have. We’ll see more on this
later.
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EXAMPLE #1
1. A Discrete Example
Consider again the RV, X = # of good components when 3 components are selected
randomly from a box containing 7 components, 3 of which are bad. We saw that X was
discrete with probability distribution given by
4!
3!
 4  3 
 x  3-x 
 = X!(4-X)! (3-X)!(3-(3-X))! , for x=0, 1, 2, 3
fX (x) =  
7!
7
3
3!4!
 
In tabular form we have
x
fX(x)
0
1/35
1
12/35
2
18/35
3
4/35
Find the mean, variance, and standard deviation of X.
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EXAMPLE #1 Again
1. A Discrete Example
Consider again the RV, X = # of good components when 3 components are selected
randomly from a box containing 7 components, 3 of which are bad. We saw that X was
discrete with probability distribution given by
In tabular form we have
x
fX(x)
0
1/35
1
12/35
2
18/35
3
4/35
Find the mean, variance, and standard deviation of X, BUT USE COMPUTATIONAL
FORMULAS!
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EXAMPLE #2
2. A Continuous Example
Let Y be a continuous RV with pdf fY(y) = 2y, for 0 < y < 1.
Use the “definitional” form to find the variance of Y.
Now use the “computational” form to find the variance.
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EXAMPLE #3
3. Another Continuous Example
Recall the RV, X = time (in hours) until the first customer enters a store after opening,
with pdf given by fX (x) = 81 , 0  x  8 .
Show that the mean and variance of X are 4 and 5 1/3, respectively.
Define a new RV, T as follows. Let T = time of day (in mins) of the first customer arrival
to the store if the store opens at 10:00am. Find the mean and variance of T.
How are T and X related?
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RULES & PROPERTIES OF EXPECTATION
Below are a series of properties/rules/results about the expectation function. They can
be used to find expected values of some functions much easier than by scratch.
1. If a is any constant, E [ a ] = a .
2. If a is any constant, E [ aX ] = a E [ X ] or more generally E [ a g(X) ] = a E [ g(X) ].
3. The expected value function is a linear function so that E [ sum of things ] = sum of
the E [ things ]. That is E [ g(X) + h(X) ] = E [ g(X) ] + E [ h(X) ].
4. If a and b are constants, then E [ a X + b ] = a E [ X ] + b .
Put another way, if Y, a new RV defined in terms of the RV X as Y = a X + b, then the
mean of Y = a * mean of X + b = a X + b.
5. If X is any RV with variance 2X and if a and b are constants, then the variance of the
Y = a X + b, is a22X.
EXAMPLE
Let D be the number of days that a particular rental car is rented. D is discrete with
sample space given by { 1, 2, 3, …, 365 } (they do not rent any car over a year!) and
probability function given by
e-5.2 5.2(d-1)
fX (x) =
, d=1, 2, 3, ....
(d-1)!
One can show that D is 5.2 days and 2D = 5.2 days2.
This car rents for $30 per day with an overhead charge of $75. If we let I be the RV
that represents the income this rental car generates, find the mean and standard
deviation of the rental income of this car.
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SUMMARY OF EXPECTED VALUES
1. E{ g(X) } =  { g(x) * [ fX(x) ] }, if X discrete and =

 g(x) * [ f (x) dx] , if X Continuous
X
-
Mean of the RV, X, is X = E{ X }
Variance of the RV, X, is σ X2 = E{ (X - X )2 } = E{ X2 } - (X )2
RULES, where a, b, and c are constants
2. aX + b = E { aX + b } = E { aX } + E{ b } =a*E { X } + b = a*X + b
2
2
2
2
σaX
+b = V { aX + b } = V { aX }= a *V { X } = a * σ X
3. aX+bY+c = E { aX+bY+c } = a E { X } + b E { Y } + c = a X + b Y + c
2
σ aX+bY+c
= V { aX+bY+c } = a2V { X } + b2V { Y } = a2 σ 2X + b2 σ 2Y , but ONLY if X and Y are
independent RV’s
Two Special Cases
4. X+Y = E { X + Y } = E { X } + E { Y } = X + Y
σ 2X + Y = V { X + Y } = V { X } + V { Y } = σ 2X + σ 2Y , but ONLY if X and Y are independent
RV
5. X -Y = E { X - Y } = E { X } - E { Y } = X - Y
σ 2X - Y = V { X - Y } = V { X } + V { Y } = σ 2X + σ 2Y , but ONLY if X and Y are independent
RV’s
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