From Algorithms to Z-Scores - matloff - University of California, Davis

10.1. SAMPLING DISTRIBUTIONS 195
What about drawing from an infinite population? This may sound odd at first, but it relates to the
fact, noted at the outset of Chapter 5, that although continuous random variables don’t really exist,
they often make a good approximation. In our human height example above, for instance, heights
do tend to follow a bell-shaped curve which which is well-approximated by a normal distributiion.
In this case, each Xi is modeled as having a continuum of possible values, corresponding to a
theoretically infinite population. Each Xi then has the same density as the population density.
10.1.2 The Sample Mean—a Random Variable
A large part of this chapter will concern the sample mean,
X = X1 + X2 + X3 + ... + Xn
n
Since X1, X2, X3, ..., Xn are random variables, X is a random variable too.
Make absolutely sure to distinguish between the sample mean and the population mean.
(10.1)
The point that X is a random variable is another simple yet crucial concept. Let’s illustrate it with
a tiny example. Suppose we have a population of three people, with heights 69, 72 and 70, and we
draw a random sample of size 2. Here X can take on six values:
69 + 69
2
= 69,
69 + 72
2
= 70.5,
69 + 70
2
= 69.5,
70 + 70
2
= 70,
70 + 72
2
= 71,
72 + 72
2
The probabilities of these values are 1/9, 2/9, 2/9, 1/9, 2/9 and 1/9, respectively. So,
pX (69) = 1
9 , p 2
X (70.5) =
9 , p 2
X (69.5) =
9 , p 1
X (70) =
9 , p 2
X (71) =
9 , p 1
X (72) =
9
Viewing it in “notebook” terms, we might have, in the first three lines:
notebook line X1 X2 X
1 70 70 70
2 69 70 69.5
3 72 70 71
Again, the point is that all of X1, X2 and X are random variables.
= 72 (10.2)
(10.3)