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Unformatted text preview: Review Notes for Loss Models 1 – ACTSC 431/831, Fall 2011 Part 1 – Random Variables and Distributional Quantities • The distribution function (df) or cumulative distribution function (cdf) and survival function (sf) of a random variable (rv) X are defined by F ( x ) = Pr { X ≤ x } and S ( x ) = Pr { X > x } = 1 F ( x ) for all x. 1. The df F ( x ) and sf S ( x ) satisfy the following four conditions: (a) 0 ≤ F ( x ) ≤ 1 (0 ≤ S ( x ) ≤ 1) for all x. (b) F ( x ) is nondecreasing ( S ( x ) is nonincreasing). (c) F ( x ) is rightcontinuous ( S ( x ) is rightcontinuous). (d) F (∞ ) = 0 and F ( ∞ ) = 1 ( S (∞ ) = 1 and S ( ∞ ) = 0). 2. Any function satisfying the above four conditions is a df (sf) of some random variable. 3. Pr { a < X ≤ b } = F ( b ) F ( a ) = S ( a ) S ( b ). 4. Pr { X = a } = Pr { X ≤ a }  Pr { X < a } = F ( a ) F ( a ). 5. If F ( x ) is continuous at x = a , then Pr { X = a } = 0. If F ( x ) is not continuous at x = a , then Pr { X = a } = F ( a ) F ( a ) is the jump size of F ( x ) at x = a . • Three types of random variables: 1. Continuous: X takes all the values in some interval such as ( a,b ), [0 , ∞ ), (∞ , ∞ ), and so on. 2. Discrete: X takes only finite values { x 1 ,...,x n } or countable values { x 1 ,x 2 ,... } . 3. Mixed: X takes all the values in some interval and some finite or countable values. • For a continuous rv X with df F ( x ): 1. Pr { X = a } = 0 for all a . 2. Pr { a < X ≤ b } = Pr { a ≤ X ≤ b } = Pr { a < X < b } = Pr { a ≤ X < b } = F ( b ) F ( a ) . • The density function or probability density function (pdf) of a continuous rv X is given by f ( x ) = F ( x ) = S ( x ) with R ∞∞ f ( y ) dy = 1 , F ( x ) = Z x∞ f ( y ) dy and S ( x ) = Z ∞ x f ( y ) dy. In addition, Pr { a < X ≤ b } = R b a f ( x ) dx. 1 • The probability function (pf) or probability mass function of a discrete rv X is denoted by p ( x j ) = Pr { X = x j } with X x j p ( x j ) = 1, F ( x ) = X x j ≤ x p ( x j ) and S ( x ) = X x j >x p ( x j ) . • The failure rate of a continuous rv X is denoted by h ( x ) = f ( x ) S ( x ) = S ( x ) S ( x ) = d dx ln S ( x ) , which is also called hazard rate or force of mortality . If X is a nonnegative continuous rv, then S ( x ) = e R x h ( t ) dt . • The k th moment of a rv X is E ( X k ) , in particular, E ( X ) is the mean or expecta tion of X ....
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 Fall '09
 laundriualt
 Probability theory, probability density function, Cumulative distribution function, Probability mass function, xj

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