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Unformatted text preview: MATH 472/567: Actuarial Theory II/Topics in Actuarial Theory I Final Exam: Formula Summary Note: This formula summary only contains formulas for material NOT covered on the two midterms. The final exam is cumulative: refer to the previous two formula summaries for key concepts for the other material. Multiple Life Functions : joint sf: s T ( x ) T ( y ) ( s, t ) = Pr [ T ( x ) > s, T ( y ) > t ] joint cdf: F T ( x ) T ( y ) ( s, t ) = Pr [ T ( x ) ≤ s, T ( y ) ≤ t ] Independence of T ( x ) and T ( y ), hereafter indicated by “IND” = ⇒ s T ( x ) T ( y ) ( s, t ) = ( s p x )( t p y ); F T ( x ) T ( y ) ( s, t ) = ( s q x )( t q y ) Joint-Life Status ( xy ): T ( xy ) = time until the first death of (x) and (y). sf: t p xy = s T ( x ) T ( y ) ( t, t ); IND = ⇒ t p xy = ( t p x )( t p y ) cdf: t q xy = 1 - t p xy ; IND = ⇒ t q xy = 1 - ( t p x )( t p y ) Force of Failure: μ xy ( t ) =- d dt t p xy t p xy ; IND = ⇒ μ xy ( t ) = μ x ( t ) + μ y ( t ) Probability Mass Function (pmf): k | q xy = k +1 q xy- k q xy = k p xy- k +1 p xy = ( k p xy )( q x + k : y + k ) This is the probability that ( xy ) remains intact for the next k years, and then fails during year ( k + 1). This is equivalent to the probability that the first death of (x) and (y) occurs in the ( k + 1)st year....
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This note was uploaded on 05/18/2011 for the course MATH 472 taught by Professor Zhu during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08