(Papers) ACET Paper May 2015 "CT3 – Probability & Mathematical Statistics"

Q. 1)

i) Given that, for any two events A and B, P [A U B] = 0.6 and P [A U ] = 0.8. Find P [A]

ii) In a certain large population, 2% of people have 6 fingers. A random sample of 400 people is chosen from this population. Calculate approximate value for the probability that 8 or more number of people in the sample have 6 fingers.

iii) Simulate three observations from exponential distribution with mean 5 using the following three random numbers from U(0,1) 0.0923, 0.8657 and 0.3494 (3)

iv) The Annual pay packages (in Lakhs of Rupees) of 9 actuarial students are given below:
2, 6, 16, 8, 5, 7, 5, 9, 3.

Present this data graphically using the Boxplot and comment on the distribution.

Q. 2)

A company sells a particular brand of a mobile at Rs. 10,000 per piece. It guarantees refund as follows. Full cost of the mobile will be refunded to the buyer if it fails during the first year, half the cost if it fails after one year but before two years, and no refund if it fails after two years. The life time of the mobile follows exponential distribution with mean two years,

i) Find the probabilities of the mobile failing in first year, in second year and after two years. (3)
ii) If the company sells 500 mobiles, calculate expected amount of refund by the company. (3)

Q. 3)

A laptop retailer procures 40%, 30%, 20% and 10% of laptops from four manufacturers A, B, C and D respectively. It is known that 15%, 10% 5% and 3% of laptops received from the respective manufacturers are faulty. A laptop is randomly chosen from the retailer and is found to be defective.

Find the probability that it is manufactured by B?

Q. 4)

A general insurance company has designed a one year motor insurance policy in such a way that if a policyholder claims for the first time, he will get Rs. 5000 and for the subsequent claims he will get Rs. 2500 each. Obviously, the policyholder will not get any amount if he has not filed any claim during a policy year.

An actuary has made an assumption that for all integers n ≥ 0, where represents the probability that the policyholder files n claims during the period

i) Find the distribution of the number of claims arising on the motor insurance policy.
ii) Find expected value and standard deviation of the claim amount for each policy.