Unformatted text preview: AMS 410 Actuarial Mathematics Fall 2009 Homework 2: Integrals + Discrete distribution Due: 10/13/2009
1. Evaluate the following integrals: (a) (b) (c) (d) (e) (f) (g)
(x3 + 2x5 + 3x10 )dx ´0 ∞ (1 + x)−5 dx ´0 ∞ x(1 + x)−5 dx ´0 −3x ∞ e dx ´1 ∞ −3x xe dx ´1 ∞ −x 2 / 2 xe dx ´−∞ ∞ max{x, 2} 1 e−x/3 dx 3 0 ´1 2. For a certain discrete random variable on the nonnegative integers, the probability function 1 satises the relationships P (0) = P (1) and P (k + 1) = k P (k), for k = 1, 2, 3,... Find P (0). 3. On a given day, each computer in a lab has at most one crash. There is a 5% chance that a computer has a crash during the day, independent of the performance of any other computers in the lab. There are 25 computers in the lab. Find the probability that on a given day, there are: (a) (b) (c) (d) exactly 3 crashes at most 3 crashes at least 3 crashes more than 3 and less than 6 crashes 4. You throw a fair die repeatedly until you get a 6. What's the probability that you need to throw more than 20 times to get 6? 5. The distribution of loss due to re damage to a warehouse is: Amount of loss 0 500 1,000 10,000 50,000 100,000 Probability 0.900 0.060 0.030 0.008 0.001 0.001 Given that a loss is greater than zero, calculate the expected amount of the loss. 6. The number of injury claims per month is modeled by a random variable N with
P (N = n) = 1 , when n ≥ 0. (n + 1)(n + 2) Determine the probability of at least one claim during a particular month, given that there have been at most four claims during that month. 1 ...
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 Fall '08
 Yang,Y
 Probability distribution, Probability theory, Probability mass function, Continuous probability distribution, Actuarial Mathematics Fall

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