Math 408, Actuarial Statistics I A.J. HildebrandDiscrete Random Variables, IINamed discrete distributions: The Big ThreeThe following is a list of essential formulas for the three most imp ortant discrete distributions:binomial, geometric, and Poisson. You are expected to know these formulas in exams, so youshould memorize them. For p.m.f.’s be sure to also memorize the range (i.e., the set of values x atwhich 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) =nxpx(1 − p)n−x(x = 0, 1, 2, . . . , n)(see back of page for definitions and properties of the binomial coefficientsnk)• Expectation and variance: µ = np, σ2= np(1 − p)• Arises as: Distribution of number of successes in success/failure trials (“Bernoulli tri-als”)2. Geometric distribution:• Parameter: p (0 < p < 1)• P.m.f.: f(x) = (1 − p)x−1p (x = 1, 2, . . .)• Expectation and variance: µ = 1/p, σ2= (1 − p)/p2• Geometric series formula:∞Xn=0rn=11 − r(|r| < 1)• Arises as: Distribution of trial at which the first success occurs in success/failure trialsequence3. Poisson distribution:• Parameter: λ > 0• P.m.f.: f(x) = e−λλxx!(x = 0, 1, 2, . . .)• Expectation and variance: µ = λ, σ2= λ• Exponential series formula:∞Xn=0λnn!= eλ• Arises as: Distribution of number of occurrences of rare events, such as accidents,insurance claims, etc.Other discrete distributionsThe following distributions are listed in the inside cover of Hogg/Tanis, but you need not memorizethe various formulas associated with these distributions. These distributions are far less importantand common than the above three, and you won’t need them for any hw/quiz/exam problems.1. Hypergeometric distribution: f(x) =(N1x)(N2n−x)(Nn), x = 0, 1, . . . , N1, n − x ≤ N22. Negative binomial distribution: f(x) =x−1r−1(1 − p)x−rpr, x = r, r + 1, . . .1Math 408, Actuarial Statistics I A.J. HildebrandBinomial coefficients• Definition: For n = 1, 2, . . . and k = 0, 1, . . . , n,nk=n!k!(n − k)!.(Note that, by definition, 0! = 1.)• Alternate notations:nCkor C(n, k)• Alternate definition:nk=n(n − 1) . . . (n − k + 1)k!.(This version is convenient for hand-calculating binomial coefficients.)• Symmetry property:nk=nn − k• Special cases:n0=nn= 1,n1=nn − 1= n• Binomial Theorem: (x + y)n=nXk=0nkxkyn−k• Binomial Theorem, special case:nXk=0nkpk(1 − p)n−k= 1• Combinatorial Interpretations:nkrepresents1. the number of ways to select k objects out of n given objects (in the sense of unorderedsamples without replacement);2. the number of k-element subsets of an n-e leme nt 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, thebinomial distribution b(n, p) is the distribution with density (p.m.f.) f(x) =nxpx(1−p)n−x,for x = 0, 1, . . . ,
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