Section 11.5 lecture notes
Math331, Fall 2008
Instructor: David Anderson
Section 11.5: Central Limit Theorem
HW: pg. 506, #’s 2, 6 (uniform), 8.
Consider X1, X2, X3, . . . , which are independent and identically distributed with mean µ
and variance σ2. In nature, it is observable that no matter what the distribution of Xiis,
X1+X2+···+Xn
looks like a normal distribution (if you back far enough away).
Example 1: Consider rolling a die 100 times (each Xiis the output from one roll) and
adding outcomes. You will get a value around 350, plus or minus some. Do this experiment
10000 times and plot the number of times you get each outcome. Will look like a bell curve.
Example 2: Go to a library and go to the stacks. Each row of books is divided into n >> 1
pieces. Let Xibe number of books on piece i. Then Pn
i=1 Xiis the number of books on
a given row. Do this for all rows of same length. You will get a plot that looks like a bell
curve. Do these bell curves have anything in common?
Consider Wn=X1+Xn+···+Xn. How far “back” should we go out to view this? What
does this mean? (scaling) Why not standardize it? Recall, to standardize we do
Wn−E[Wn]
σWn
.
This has mean zero (shift over to zero) and variance 1. This seems like right way to “back
off.” So,
E[Wn] = E[X1+Xn+···+Xn] = nµ.
σWn=pV ar(X1+···+Xn) = pV ar(X1) + ···+V ar(Xn) = √nσ2=σ√n.
So, what does Wn−nµ
σ√n=X1+···+Xn−nµ
σ√n
look like for large n?
Theorem 1 (Central Limit Theorem).Let X1, X2,... be a sequence of independent and
identically distributed random variables, each with expectation µand variance σ2. Then the
distribution of
Zn=X1+···+Xn−nµ
σ√n
converges to the distribution of a standard normal random variable. That is,
lim
n→∞
P(Zn≤t) = lim
n→∞
PX1+···+Xn−nµ
σ√n≤t
=1
√2πZt
−∞
e−x2/2dx.
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