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The Binomial Theorem The Binomial Theorem states that the following binomial formula is valid for all positive nn 1 n2 2 nn 1n 2 n3 3 integer values of n: a b a n na n1b a b a b ... b n . The n 2! 3! theorem makes it possible to expand finite binomials to any given power without direct n! n multiplication. The general formula is a b a nk b k , which expresses a in descending n k 0 k n! n powers and a b n a k b k n expresses a in ascending powers. The Theorem also provides k 0 k a method of determining the binomial coefficients of each term without having to multiply an expansion out. The n represents the power of the formula and the k represents the power to n n! which b is raised. Thus, k k! n k ! n C k is the combination formula. The n! is called the „n factorial‟ and represents the product of the first n positive integers, n! = n (n-1) (n-2) … (2) (1). For example, in the expansion of a b the coefficient of the third term, a 3b 2 can be found 5 5! 5 4 3 2 1 120 by using 10 2!5 2! 2 1 3 2 1 12 There are certain properties that the expansions have and are assumed valid where the exponent of the binomial, (a + b)n, is any positive integer. The properties state: 1. The number of terms, in the series, equals the binomial exponent plus one 2. The first term and last term of the series is an and bn 3. Progressing from the first term to the last, the exponent of a decreases by one from term to term, the exponent of b increases by one from term to term, and the sum of the exponents of a and b in each term is n 4. If the coefficient of any term is multiplied by the exponent of a in that term, and this product is divided by the number of that term, the coefficient of the next term is obtained 5. The coefficients of terms equidistant from the ends are equal. Property 5 Property 2, n Property 2, n For example: Property 3 3+2=5 a b5 a 5 5a b 10a 3b 2 10a 2 b 3 5ab 4 b 5 4 5x4 2nd term 2 Property 4 One role that the Binomial Theorem plays in probability and statistics is it is used in the process of finding discrete binomial distributions. The Binomial Theorem provides information of what happens when a binomial expression is raised to a power that leads to a binomial expansion. A binomial distribution is described by a formula related to the binomial expansion‟s binomial coefficient that gives the number of trials and successes. This type of distribution is used in the Bernoulli trials, which uses a fixed number of trials with fixed probability of success on each trial. A random variable n has a Bernoulli distribution with parameter p if it can assume a value of n=1 (success) with a probability of p and the value of n=0 (failure) with a probability p n 1 of (1-p). Therefore the probability function is p(n) . Consider a statistical 1 p n 0 experiment where a success occurs with probability p and a failure occurs with probability q=1- p. If the experiment is repeated for n times, with each independent trial, then the random variable k, whose value is the number of successes in the n trials, has a binomial distribution with n parameters p and n and has the following probability function: p(k ) p k 1 p nk for k = 0, k 1, 2, …, n. For example, the probability that in n throws of a die a throw of 6 will occur k times k nk n 1 5 is pk k 6 . A die has 6 faces, so getting a roll of 6 is 1/6th of a chance, and 6 getting a roll other than 6 is 5/6th of a chance. Let n = 20 and k = 5, then 20 5 20 1 5 5 20! 2432902008 176640000 Pk 5 6 5!20 5! .000128 .064866 1569209241 60000 .0000083 .129 6 So the probability of getting a 6, 5 times in 20 rolls is .129. The Binomial Theorem‟ binomial expansion is also used to describe the distribution of values about a mean, for example, the average of the number of successes will approach a mean value µ given by the probability for success of each item p times the number of items. The mean µ of the binomial distribution is evaluated by combining the definition of µ with the function that n n! defines the probability, yielding: k p k (1 p) nk np . When p is the k 0 k!(n k )! probability of success on each trial, the expected number of successes in n trials is np. The third role the Theorem plays is in finding the variance of k. The variance is a measure of the spread of the distribution about the mean and is defined by: n n! 2 (k ) 2 p k (1 p) nk np(1 p) . This too is found by using the binomial k 0 k!(n k )! expansion. The Binomial Theorem‟s role in probability and statistics provides the binomial coefficients in many applications to find various aspects of observations, such as distributions based on permutations and combinations. The Theorem also provides a way to expand binomial expressions used in binomial distributions such as those used in Bernoulli Trails. These binomial expansions are also used in finding the mean and variance of distribution. REFERENCES: Math Advantage: Statistics. CD-Rom, 1997. Ace Research, Inc. PlanetMath.org. <http://planetmath.org/?op=getobj&from=objects&id=338>. Shapiro, Perry. “Introduction to Probability and Statistics for Economists.” Sept. 2005. <http://www.econ.ucsb.edu/~pxshap/econ241a/Notes%20for%202005.pdf>. Wagner, Carl. “Choice, Chance, and Inference.” An Introduction to Combinatorics, Probability and Statistics. <http://www.math.utk.edu/~wagner/papers/book.pdf> Weisstein, Eric. "Bernoulli Distribution." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/BernoulliDistribution.html> Wosik, Ewa. BERNOULLI TRIALS and the BINOMIAL DISTRIBUTION. Connexions. 26 Nov. 2005. <http://cnx.rice.edu/content/m13123/1.1/>. Virtual Laboratories. “Bernoulli Trials.” <http://www.ds.unifi.it/VL/VL_EN/bernoulli/bernoulli1.html>.
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