AQ017-3-1-APROM Advanced Probability Models Assignment

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AQ017-3-1-APROM Advanced Probability Models Assignment - Asia Pacific University of Technology & Innovation, Malaysia

Instruction - Answer ANY 4 Questions.

QUESTION 1 - (a) X is a discrete random variable with cumulative distribution function

Calculate the variance of X.

(b) The lifetime of a light bulb has density function

Calculate the mode of this distribution.

(c) An insurance policy covers losses due to theft, with a deductible of 3. Theft losses are uniformly distributed on [0, 10]. Determine the moment generating function, M(t), for t ≠ 0, of the claim payment on a theft.

(d) Let X₁, X₂, X₃ be a random sample from a discrete distribution with PMF

Determine the moment generating function, M(t), of Y = X₁ + X₂ + X₃.

QUESTION 2 - (a) The time, T, to failure of a device is exponentially distributed with mean 3 years. Since the device will not be monitored during its first two years of service, the time to discovery of its failure is X = max(T, 2). Find E(X).

(b) The distribution of the size of claims paid under an insurance policy has PDF

Where k and m are constants, k > 0 and m > 0. For a randomly selected claim, the probability that the size of the claim is less than 3.75 is 0.4871. Find the probability that the size of a randomly selected claim is greater than 4.

(c) In a shipment of 20 packages, 7 packages are damaged. The packages are randomly inspected, one at a time, without replacement, until the fourth damaged package is discovered. Find the probability that exactly 12 packages are inspected.

(d) Given that P(A U B) = 0.7 and P(A U B') = 0.9, find P(A).

QUESTION 3 - (a) X and Y are two independent discrete random variables with PMF defined in the tables below.

x	0	1	2	3
f_X(x)	0.1	0.2	0.3	0.4

y	0	1	2
f_Y(x)	0.25	0.4	0.35

If Z = X × Y, find E(Z).

(b) Let X and Y be continuous random variables with PDF

Find E(X | Y = y).

(c) For a certain insurance company, 10% of its policies are Type A, 50% are Type B, and 40% are Type C. The annual number of claims for an individual Type A, Type B, and Type C policy follow Poisson distributions with respective means 1, 2, and 10. Let X be the annual number of claims of a randomly selected policy. Calculate the variance of X.

(d) Suppose an experiment consists of tossing a fair coin until four heads occur. What is the probability that the experiment ends after exactly seven flips of the coin with a head on the sixth toss?

QUESTION 4 - (a) Let X and Y be random variables with joint moment generating function

M_XY(t₁, t₂) = 0.3 + 0.1e^t_1 + 0.2e^t_2 + 0.4e^t_1+t_2

Find Var(2X - Y).

(b) Let X and Y be continuous variables with joint density function

Find F(4, 5).

What is P(X₁ < X₂ < X₃)?

(d) Two numbers are chosen independently and at random from the interval (0, 1). What is the probability that the two numbers differ by more than 0.5?

QUESTION 5 - (a) An insurance policy reimburses dental expenses, X, up to a maximum benefit of 250. The PDF for X is

where c is a constant. Calculate the median benefit for this policy.

(b) An auto insurance policy has a deductible of 1 and a maximum claim payment of 5. Auto loss amounts follow an exponential distribution with mean 2. Calculate the expected claim payment made for an auto loss.

(c) A family buys two policies from the same insurance company. Losses under the two policies are independent and have continuous uniform distributions on the interval from 0 to 10. One policy has a deductible of 1 and the other has a deductible of 2. The family experiences exactly one loss under each policy. Calculate the probability that the total benefit paid to the family does not exceed 5.

(d) An automobile insurance company issues a one-year policy with a deductible of 500. The probability is 0.8 that the insured automobile has no accident and 0 that the automobile has more than one accident. If there is an accident, the loss before application of the deductible is exponentially distributed with mean 3000. Calculate the 95^th percentile of the insurance company payout on this policy.

Attachment:- Advanced Probability Models Assignment File.rar

Reference no: EM132827043

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