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a_ct10l02 - 1 ACTL1001 Games of Chance(2 Outline 1...

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Unformatted text preview: 1 ACTL1001 Games of Chance (2) Outline: 1. Independence 2. Distributions in Actuarial Science (ctd) 3. Probability of Ruin 4. Simulation Reading (Req) Sherris, 2.3-2.7 2 Independence & Trials or events are independent if the occurrence of one does not the affect the probability of occurrence of any of the others. & The probability that a number of independent events occur is just the product of the probabilities that each event occurs. 3 Geometric & The Geometric distribution gives the probability of the number of trials to the first success where the probability of success on any trial is p . & Geometric ( p ) Pr ( X = x ) = (1 ¡ p ) x ¡ 1 p x = 1 ; 2 ::: E [ X ] = 1 p V ar [ X ] = 1 p & 1 p ¡ 1 ¡ 4 Example & Passing or gaining exemption from actuarial professional examinations is notoriously difficult. & Assume that the probability of passing or gaining an exemption from a particular actuarial professional examination is : 6 , and that the chance per attempt is the same and are independent. & Calculate the probability that it will take at least 3 attempts to pass or gain an exemp- tion from this examination. 5 Exponential & The exponential distribution is used for the probability distribution of the time until an event occurs where the probability does not depend on elapsed time & Exponential distribution often used to model the time of survival Exponential ( & ) f ( x ) = e ¡ x & & x ¢ and otherwise F ( x ) = Z x ¡1 f ( t ) dt = 1 ¡ e ¡ x & x ¢ E [ X ] = & V ar [ X ] = & 2 6 Normal Distribution & The normal distribution is the “bell” shaped symmetric distribution that typically re-...
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a_ct10l02 - 1 ACTL1001 Games of Chance(2 Outline 1...

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