Central Limit Theorems used in Math 201 THEOREMS 1) Theorem 6.1 in your text book. 2) Theorem 6.2 in your text book. 3) Theorem for sums of independent and identically distributed random variables: A large number of independent and identically distributed random variables is approximately normally distributed. approximately has a normal distribution with mean and variance That is, where each has mean and variance , i=1,…,n , as n goes to infinity. For practical purposes, n greater or equal to 30 will suffice. 4) If each has a discrete distribution, then you should do a continuity correction for the sum as explained below. Or, you can opt for 2c) below, although the result will have more error. CONTINUITY CORRECTION (Please don’t read this before you have studied the material covered so far) The correction for continuity , mentioned in section 5.5 of your book, is not only limited to approximating a binomial distribution with a normal distribution. It is used whenever we are approximating a discrete distribution with a continuous distribution. The two general cases that we encountered in this course are SUM and AVERAGE. 1) If we have a sum of N discrete variables, and N is greater than 30, then we can assume that the random variable representing this sum, SUM, is approximately normally distributed (Central Limit Theorem). Yet since SUM is a sum of discrete random variables, all of which can have only discrete values, SUM itself can only have discrete values. So there are really two random variables called SUM, one discrete, and one continuous. Lets call them dSUM and cSUM. Suppose we are interested in the probability that the dSUM is between the discrete values of m and n, i.e. Pr(m≤dSUM≤n). This is approximately equal to the probability that cSUM is between two continuous values, mc and nc. Pr(m ≤ dSUM ≤ n) ≈ Pr(mc ≤ cSUM ≤ nc) What are these values mc and nc ? Suppose that the discrete value for dSUM before m is , and the discrete value for dSUM after n is . Then mc = /2, and nc = /2 1, and 1 . Hence, mc = m ½ and nc= n ½ . Usually, 2a) The discrete random variable AVERAGE is defined as dSUM/N. The probability that AVERAGE is between two values a and b can thus be converted to the probability that dSUM is between the values a*N and b*N, which are discrete by nature! Then, the method outlined in part (1) can be used. 2b) If you want to work directly with the random variable AVERAGE, then you may use the values a and b DIRECTLY in the approximation WITHOUT any continuity correction! (The approximation error will increase in this case): 2c) You can also use AVERAGE by converting Pr(m ≤ dSUM ≤ n) to Pr(m/N ≤ AVERAGE ≤ n/N) and use the central limit theorem for averages and not do the continuity correction (again, the approximation error will increase in this case).