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What is the most important probability distribution? Why do you choose that distribution as most important?
The proofs in this section are drawn from Chapter 8, ”Limit Theorems”, A First Course in Probability, by Sheldon Ross, Macmillan, 1976. Further examples and considerations come from Heads or Tails: An Introduction to Limit Theorems in Probability, by Emmanuel Lesigne, American Mathematical Society, Chapter 7, pages 29-74; An Introduction to Probability Theory and Its Applications, Volume I, second edition, William Feller, J. Wiley and Sons, 1957, Chapter VII, and Dicing with Death: Chance, Health, and Risk by Stephen Senn, Cambridge University Press, Cambridge, 2003.
Lemma 1 Let X_{1},X_{2},... be a sequence of random variables having cumulative distribution functions F_{Xn} and moment generating functions . Let X be a random variable having cumulative distribution function F_{X} and moment generating function . If , for all t, then for all t at which is continuous.
We say that the sequence X_{i} converges in distribution to X and we write
This lemma is useful because it is fairly routine to determine the pointwise limit of a sequence of functions using ideas from calculus. It is usually much easier to check the pointwise convergence of the moment generating functions than it is to check the convergence in distribution of the corresponding sequence of random variables.
We won’t prove this lemma, since it would take us too far afield into the theory of moment generating functions and corresponding distribution theorems. However, the proof is a fairly routine application of ideas from the mathematical theory of real analysis.
Here’s a simple representative example of using the convergence of the moment generating function to prove a useful result. We will prove a version of the Weak Law of Large numbers that does not require the variance of the sequence of independent, identically distributed random variables.
Theorem 2 (Weak Law of Large Numbers) Let X_{1},...,X_{n} be independent, identically distributed random variables each with mean and such that is finite. Let S_{n} = X_{1}++X_{n}. Then S_{n}/n converges in probability to . That is:
Proof: If we denote the moment generating function of X by , then the moment generating function of
is The existence of the first moment assures us that is differentiable at 0 with a derivative equal to . Therefore, by Taylor expansion with remainder
where r(t/n) is a remainder function such that
Then we need to consider
Taking the logarithm of and using L’Hospital’s Rule, we see that
But this last expression is the moment generating function of the (degenerate) point mass distribution concentrated at . Hence,
Q.E.D.
Theorem 3 (Central Limit Theorem) Let X_{1},...X_{n} be independent, identically distributed random variables each with mean and variance . Consider Let
Proof: Assume at first that and . Assume also that the moment generating function of the X_{i}, (which are identically distributed, so there is only one m.g.f) is , exists and is everywhere finite. Then the m.g.f of X_{i}/ is
To handle the general case, consider the standardized random variables , each of which now has mean 0 and variance 1 and apply the result. Q.E.D.
The first version of the central limit theorem was proved by DeMoivre around 1733 for the special case when the X_{i} are binomial random variables with p = 1/2 = q. This proof was subsequently extended by Laplace to the case of arbitrary pq. Laplace also discovered the more general form of the Central Limit Theorem presented here. His proof however was not completely rigorous, and in fact, cannot be made completely rigorous. A truly rigorous proof of the Central Limit Theorem was first presented by the Russian mathematician Liapunov in 1901-1902. As a result, the Central Limit Theorem (or a slightly stronger version of the CLT) is occasionally referred to as Liapunov’s theorem. A theorem with weaker hypotheses but with equally strong conclusion is Lindeberg’s Theorem of 1922. It says that the sequence of random variables need not be identically distributed, but instead need only have zero means, and the individual variances are small compared to their sum.
The statement of the Central Limit Theorem does not say how good the approximation is. In general the approximation given by the Central Limit Theorem applied to a sequence of Bernoulli random trials or equivalently to a binomial random variable is acceptable when np(1 - p) > 18. The normal approximation to a binomial deteriorates as the interval (a,b) over which the probability is computed moves away from the binomial’s mean value np.
The Berry-Esséen Theorem gives an explicit bound: For independent, identically distributed random variables X_{i} with , , and , then
We expect the normal distribution to arise whenever the outcome of a situation, results from numerous small additive effects, with no single or small group of effects dominant. Here is an illustration of that principle.
This illustration is adapted from Dicing with Death: Chance, Health, and Risk by Stephen Senn, Cambridge University Press, Cambridge, 2003.
Consider the following data from an American study called the National Longitudinal Survey of Youth (NLSY). This study originally obtained a sample of over 12,000 respondents aged 14-21 years in 1979. By 1994, the respondents were aged 29-36 years and had 15,000 children among them. Of the respondents 2,444 had exactly two children. In these 2,444 families, the distribution of children was boy-boy: 582; girl-girl 530, boy-girl 666, and girl-boy 666. It appears that the distribution of girl-girl family sequences is low compared to the other combinations, our intuition tells us that all combinations are equally likely and should appear in roughly equal proportions. We will assess this intuition with the Central Limit Theorem.
Consider a sequence of 2,444 trials with each of the two-child families. Let X_{i} = 1 (success) if the two-child family is girl-girl, and X_{i} = 0 (failure) if the two-child family is otherwise. We are interested in the probability distribution of
In particular, we are interested in the probability , that is , what is the probability of seeing as few as 530 girl-girl families or even fewer in a sample of 2444 families? We can use the Central Limit Theorem to estimate this probability.
We are assuming the family “success” variables X_{i} are independent, and identically distributed, a reasonable but arguable assumption. Nevertheless, without this assumption, we cannot justify the use of the Central Limit Theorem, so we adopt the assumption. Then and Var[X_{i}] = (1/4)(3/4) = 3/16 so Hence
We expect the normal distribution to arise whenever the outcome of a situation, results from numerous small additive effects, with no single or small group of effects dominant. Here is another illustration of that principle.
The following is adapted from An Introduction to Probability Theory and Its Applications, Volume I, second edition, William Feller, J. Wiley and Sons, 1957, Chapter VII.3(e), page 175.
The Central Limit Theorem can be used to assess risk. Two large banks compete for customers to take out loans. The banks have comparable offerings. Assume that each bank has a certain amount of funds available for loans to customers. Any customers seeking a loan beyond the available funds will cost the bank, either as a lost opportunity cost, or because the bank itself has to borrow to secure the funds to loan to the customer. If too few customers take out loans, then that also costs the bank, since now the bank has unused funds.
We create a simple mathematical model of this situation. We suppose that the loans are all of equal size and for definiteness each bank has funds available for a certain number (to be determined) of these loans. Then suppose n customers select a bank independently and at random. Let X_{i} = 1 if customer i select bank H with probability 1/2 and X_{i} = 0 if customers select bank T, also with probability 1/2. Then is the number of loans from bank H to customers. Now there is some positive probability that more customers will turn up than can be accommodated. We can approximate this probability with the Central Limit Theorem:
so if n = 1000, then s = 537 will suffice. If both banks assume the same risk of sellout at 0.01, then each will have 537 for a total of 1074 loans, of which 74 will be unused. In the same way, if the bank is willing to assume a risk of 0.20, i.e. having enough loans in 80 of 100 cases, then they would need funds for 514 loans, and if they want to have sufficient seats in 999 out of 1000 cases, they should have 549 loans available.
Now the possibilities for generalization and extension are apparent. A first generalization would be allow the loan amounts to be random with some distribution. Still we could apply the Central Limit Theorem to approximate the demand on available funds. Second, the cost of either unused funds or lost business could be multiplied by the chance of occurring. The total of the products would be an expected cost, which could then be minimized.
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Steve Dunbar's Home Page, http://www.math.unl.edu/~sdunbar1Last modified: Tuesday, 09-Oct-2007 06:33:51 CDT