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([pic])?

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.

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 |0.5 |

|2 |0.25 |

|3 |0.125 |

|4 |0.0625 |

|5 |0.03125 |

|6 |0.015625 |

|7 |0.007813 |

|8 |0.003906 |

|9 |0.001953 |

|10 |0.000977 |

|11 |0.000488 |

|12 |0.000244 |

|13 |0.000122 |

|14 |6.1E-05 |

|15 |3.05E-05 |

|16 |1.53E-05 |

|17 |7.63E-06 |

|18 |3.81E-06 |

|19 |1.91E-06 |

|20 |9.54E-07 |

|21 |4.77E-07 |

|22 |2.38E-07 |

|23 |1.19E-07 |

|24 |5.96E-08 |

|25 |2.98E-08 |

|26 |1.49E-08 |

|27 |7.45E-09 |

|28 |3.73E-09 |

|29 |1.86E-09 |

|30 |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!

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!!!

Below are examples of four VERY different continuous distributions, as denoted by their pdf’s.

|[pic] |[pic] |

|[pic] |[pic] |

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.

Defn: Let X be any RV with probability distribution, fX(x). The Expected Value Function is defined to be

E [ g(X) ] = [pic]

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.

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

[pic]

In tabular form we have

|x |fX(x) | | |

|0 | | | |

|1 | | | |

|2 | | | |

|3 | | | |

| | | | |

Find the expected value of X.

Find the expected value of[pic].

EXAMPLE #2

Devore/Farnum—X = amount of gravel (100tons) sold per week by supply company.

[pic]

If the company makes $2.50 profit on each ton sold, what is the company’s expected/average/mean profit per week?

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 σ2X and defined as

σ2X = 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

σ2X = 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 σX and is simply the positive square root of the variance σX = √σ2X .

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.

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

[pic]

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.

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!

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.

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[pic].

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?

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 σ2X and if a and b are constants, then the variance of the

Y = a X + b, is a2σ2X.

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

[pic]

One can show that (D is 5.2 days and σ2D = 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.

SUMMARY OF EXPECTED VALUES

1. E{ g(X) } = ( { g(x) * [ fX(x) ] }, if X discrete and =[pic], if X Continuous

Mean of the RV, X, is μX = E{ X }

Variance of the RV, X, is [pic]= E{ (X - μX )2 } = E{ X2 } - (μX )2

RULES, where a, b, and c are constants

2. μaX + b = E { aX + b } = E { aX } + E{ b } =a*E { X } + b = a*μX + b

[pic]= V { aX + b } = V { aX }= a2*V { X } = a2*[pic]

3. μaX+bY+c = E { aX+bY+c } = a E { X } + b E { Y } + c = a μX + b μY + c

[pic]= V { aX+bY+c } = a2V { X } + b2V { Y } = a2[pic] + b2[pic], but ONLY if X and Y are independent RV’s

Two Special Cases

4. μX+Y = E { X + Y } = E { X } + E { Y } = μX + μY

[pic]= V { X + Y } = V { X } + V { Y } = [pic] + [pic], but ONLY if X and Y are independent RV

5. μX -Y = E { X - Y } = E { X } - E { Y } = μX - μY

[pic]= V { X - Y } = V { X } + V { Y } = [pic] + [pic], but ONLY if X and Y are independent RV’s

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