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Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009 Part II Univariate Distribution updated on 10/13/2009 1 Discrete Random Variables, I 1.1 Terminology random variable : a quantity X whose value depends on some random event. The space (or range ) of X : the set S of possible values of X . If S is nite or countable , the random variable is called discrete . 1.2 General formulas 1. Probability mass function (p.m.f.) (also called a discrete density function , or discrete distribution ) De nition and notation: f ( x ) = P ( X = x ) for all x S Properties: (1) f ( x ) , (2) x S f ( x ) = 1 (the summation being over the range of f ) Uniform distribution on a set S : Each of the values x S has the same probability, i.e., f ( x ) = 1 /n for each value x , where n is the number of values. 2. Cumulative distribution function (c.d.f.) De nition and notation: F ( x ) = P ( X x ) = w x f ( w ) . Properties: (1) F ( x ) is non-decreasing, (2) F ( x ) 1 , (3) lim x F ( x ) = 1 , (4) lim x - F ( x ) = 0 . Survival function: S ( x ) = 1- F ( x ) = P ( X > x ) . 3. Expectation (mean) De nition and notation: = E ( X ) = x S xf ( x ) Properties: E ( c ) = c , E ( cX ) = cE ( X ) , E ( X + Y ) = E ( X ) + E ( Y ) , E ( X Y ) 6 = E ( X ) E ( Y ) . Expectation of a function of X : E ( u ( X )) = x S u ( x ) f ( x ) . 4. Probability calculation: P ( A ) = x A f ( x ) . 5. Variance De nition and notation: 2 = V ar ( X ) = E (( X- ) 2 ) = E ( X 2 )- E ( X ) 2 Properties: V ar ( c ) = 0 , V ar ( cX ) = c 2 V ar ( X ) , V ar ( X + c ) = V ar ( X ) . 1 Standard deviation: = p V ar ( X ) . 6. Moment generating function (m.g.f.): De nition and notation: M ( t ) = E ( e tX ) = x S e tx f ( x ) . Properties: The derivatives of M ( t ) at 0 generate the moments of X : M (0) = E ( X ) , M 00 (0) = E ( X 2 ) , M 000 (0) = E ( X 3 ) , etc. 1.3 Tips Remember to specify the set of values of a p.m.f. Distinguish between X (a random variable) and x (the value of a random variable) Expectation, variance are numbers, not functions 2 Discrete Random Variables, II 2.1 The Big Three 1. Binomial distribution Distribution of number of successes in success/failure trials ( Bernoulli trials )...
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