Chapter 10
The Expected Value and
Standard Error
This chapter introduces us to the very important concepts of random pro
cesses and their sampling distributions. To keep ideas manageable, it is
best motivated by way of example.
10.1 Expected value
Assume 100 draws are made randomly with replacement from a box con
taining 4 tickets labelled 1,1,1, and 5 as follows:
1
1
1
1
5
What would be the expected sum of the 100 draws? We can expect the
ticket labelled 1 to be picked 75 times and the ticket labelled 5 to be picked
25 times on average. So the sum would be, on average, 1
×
75+5
×
25 = 200
.
This average, of the sum of 100 draws, is known as the
expected value
.
1
This is the ﬁrst example in which we use what is called a box model. Note that this
kinds of model will be used throughout the book.
1
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CHAPTER 10. THE EXPECTED VALUE AND STANDARD ERROR
A quick way to get the expected value, usually denoted EV, for the sum
of draws made at random with replacement from a box equals:
EV
= Average of box
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 Spring '10
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 Normal Distribution, Standard Error, Variance, draws

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