Math 408 Actuarial Statistics I A J Hildebrand Discrete Random Variables II Named discrete distributions The Big Three The following is a list of essential formulas for the three most important discrete distributions binomial geometric and Poisson You are expected to know these formulas in exams so you should memorize them For p m f s be sure to also memorize the range i e the set of values x at which f x is defined along with the formula for f x 1 Binomial distribution b n p Parameters n positive integer p 0 p 1 P m f f x nx px 1 p n x x 0 1 2 n see back of page for definitions and properties of the binomial coefficients n k 2 Expectation and variance np np 1 p Arises as Distribution of number of successes in success failure trials Bernoulli trials 2 Geometric distribution Parameter p 0 p 1 P m f f x 1 p x 1 p x 1 2 Expectation and variance 1 p 2 1 p p2 X 1 rn Geometric series formula r 1 1 r n 0 Arises as Distribution of trial at which the first success occurs in success failure trial sequence 3 Poisson distribution Parameter 0 x P m f f x e x x 0 1 2 Expectation and variance 2 X n Exponential series formula e n n 0 Arises as Distribution of number of occurrences of rare events such as accidents insurance claims etc Other discrete distributions The following distributions are listed in the inside cover of Hogg Tanis but you need not memorize the various formulas associated with these distributions These distributions are far less important and common than the above three and you won t need them for any hw quiz exam problems N2 Nx1 n x x 0 1 N1 n x N2 Nn x r r 2 Negative binomial distribution f x x 1 p x r r 1 r 1 1 p 1 Hypergeometric distribution f x 1 Math 408 Actuarial Statistics I A J Hildebrand Binomial coefficients n n Definition For n 1 2 and k 0 1 n k k n k Note that by definition 0 1 Alternate notations n Ck or C n k n n n 1 n k 1 Alternate definition k k This version is convenient for hand calculating binomial coefficients n n Symmetry property k n k n n n n Special cases 1 n 0 n 1 n 1 n X n k n k n Binomial Theorem x y x y k k 0 Binomial Theorem special case n X n k 0 k pk 1 p n k 1 n Combinatorial Interpretations represents k 1 the number of ways to select k objects out of n given objects in the sense of unordered samples without replacement 2 the number of k element subsets of an n element set 3 the number of n letter HT sequences with exactly k H s and n k T s Binomial distribution Given a positive integer n and a number p with 0 p 1 the n x binomial distribution b n p is the distribution with density p m f f x p 1 p n x x for x 0 1 n 2
View Full Document
Unlocking...