Notes-Part1

# Notes-Part1 - Review Notes for Loss Models 1 ACTSC 431/831...

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Part 1 – Random Variables and Distributional Quantities 1. 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. (a) The df F ( x ) and sf S ( x ) satisfy the following four conditions: (1) 0 F ( x ) 1 for all x (0 S ( x ) 1 for all x ). (2) F ( x ) is non-decreasing ( S ( x ) is non-increasing). (3) F ( x ) is right-continuous ( S ( x ) is right-continuous). (4) F ( -∞ ) = 0 and F ( ) = 1 ( S ( -∞ ) = 1 and S ( ) = 0). (b) Any function satisfying the above four conditions is a df (sf) of some random variable. (c) Pr { a < X b } = F ( b ) - F ( a ) = S ( a ) - S ( b ). (d) Pr { X = a } = Pr { X a } - Pr { X < a } = F ( a ) - F ( a - ). (e) 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 . 2. Three types of random variables: (a) Continuous: X takes all the values in some interval such as ( a, b ), [0 , ), ( -∞ , ), and so on. (b) Discrete: X takes only ﬁnite values { x 1 , ..., x n } or countable values { x 1 , x 2 , ... } . (c) Mixed: X takes all the values in some interval and some ﬁnite or countable values. 3. For a continuous rv X with df F ( x ): (a) Pr { X = a } = 0 for all a . (b) Pr { a < X b } = Pr { a X b } = Pr { a < X < b } = Pr { a X < b } = F ( b ) - F ( a ) . 4. The density function or probability density function (pdf) of a continuous rv X is given by f ( x ) = F 0 ( x ) = - S 0 ( x ) with R -∞ f ( y ) dy = 1 , F ( x ) = Z x -∞ f ( y ) dy and S ( x ) = Z x f ( y ) dy. In addition,

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• Fall '09
• david
• Probability theory, probability density function, Cumulative distribution function, Probability mass function, xj

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Notes-Part1 - Review Notes for Loss Models 1 ACTSC 431/831...

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