A Proof of the Central Limit Theorem

Preliminaries: Two Definitions and Two Theorems

Definition One: The rth moment of a random variable X about the mean is

[Note: It follows that and.]

Thus, we have

(discrete variable)

(continuous variable)

Definition Two: The moment-generating function (MGF) of X is

Thus, we have

(discrete case)

(continuous case)

[Note: The name moment-generating function comes from the fact that

which shows the moments about the origin [] are generated as the coefficients in this expansion in.]

Proof of Central Limit TheoremPage Two

Theorem One: The MGF for the general normal distribution is

Proof: By definition, we have

Letting ( / = so that and, this becomes

Now letting, we have

Corollary: The MGF for the standard normal distribution is

Theorem Two (Uniqueness Theorem): Two random variables have the same probability distribution if and only if their moment-generating functions are identical.

We are now ready to prove the Central Limit Theorem.

Proof of Central Limit Theorem (CLT)

Theorem (CLT): Let be independent random variables that are identically distributed (that is, all have the same probability mass function in the discrete case or probability density function in the continuous case) and have finite mean and variance . Then if (n = 1, 2, …),

that is, the random variable ( ), which is the standardized variable corresponding to , is asymptotically normal.

Note: Since ( ) = ) = , the more familiar form of the CLT follows as a corollary.

Proof of Central Limit TheoremPage Three

Proof: For …, we have Now each have mean and variance Thus,

and, because the are independent,

It follows from this that the standardized random variable corresponding to is

The MGF for is thus

]

]]]

[The last step is a result of the being identically distributed and the preceding step because the are independent.]

Now, by a Taylor series expansion, we have

Thus, we have

But the limit of this as is, which is the MGF of the standard normal distribution. Hence, by Theorem Two above, the required result follows.

Proof of Central Limit Theorem Page Four

Corollary: Suppose the population from which samples are taken has some probability distribution with mean and variance . Then the standardized variable associated with , given by

is asymptotically normal, that is

Note: The above proof of the CLT, slightly edited, was given by the late Murray R. Spiegel of RPI in his 1975 Schaum’s Outline Series book Probability and Statistics in Problem 4.25, which appears as such in the current Third Edition of the book, with John J. Schiller and R. Alu Srinivasan of Temple University now as coauthors.