(Answered) Quesiton 1 johns annual income in dollars follows a

Statistics

Quesiton 1: John's annual income (in dollars) follows a two-pointmixture distribution, where 30% has a Pareto distribution with mean 1,000 and variance 3,000,000, and 70% has an Exponential distribution with mean 60,000.

(a) Calculate John's expected annual income.

(b) Calculate the variance of his annual income.

Quesiton 2: The number of accidents in a highway during a month follows a Poisson distribution with mean , where follows an Exponential distribution with mean 5. Calculate the probability of only having one accident this month.

Quesiton 3: The future lifetime of a random individual (in years) has constant force of mortality , where has a Uniform distribution on the interval [0:025; 0:04].

(a) Calculate the probability that a random individual does not survive 30 years from today.

(b) Using the normal approximation, calculate the probability that a random individual does not survive 30 years. Compare to above.

(c) An individual buys a life insurance policy for 1 million dollars, and keeps it until her death. Calculate the actuarial present value of the policy's death bene?t. Consider the annual continuously compounded interest rate to be 3%.

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