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RESEARCH SCHOOL OF FINANCE, ACTUARIAL STUDIES

AND STATISTICS

INTRODUCTORY MATHEMATICAL STATISTICS PRINCIPLES OF MATHEMATICAL STATISTICS

(STAT2001/6039)

Assignment 2 (2018 Semester 1)

Your solutions to this assignment should be placed in the appropriate box in the RSFAS School foyer by the due time and date (as provided in the Course Outline and on Wattle). Attach a cover sheet (as provided on Wattle) which has your ANU ID number. The assignment is out of 100 and is worth 10% of your overall course mark. Each problem has equally weighted parts. Show all relevant working and draw a box around each required result. Unless otherwise indicated, present final numerical solutions as exact simple fractions or correct to five significant digits (e.g. 0.024380).

Problem 1 [30 marks]

An insurance company offers a particular kind of policy whereby the insured person pays a premium of $200 and thereby becomes eligible to receive compensation in the event of a particular kind of accident whose probability of occurring during the policy period is 100p%. The policy will pay the costs arising from the accident, assumed to be exponential with mean $900, but only that part of the costs in excess of $500, and only up to a maximum payout of $4,000. Assume that at most one accident can occur.

(a) Find a, the probability that a policy holder will incur an overall loss over the policy period of more than $2,000, taking into account the premium and costs associated with the potential accident. Also, find ß, the probability that a person without the policy (but subject to the same risks as a policy holder) will incur an overall loss over the same period of more than $2,000, taking into account the costs associated with the potential accident. Express a and ß as functions of p. Then evaluate a and ß when p = 0.03.

(b) Find m, the expected value of the overall loss which a policy holder will incur over the policy period, taking into account the premium and the costs associated with the potential accident. Express m as a function of p. Then evaluate m when p = 0.03.

(c) Find g, the expected value of the overall profit which the insurance company will make from a single policy, taking into account the premium and the costs associated with the potential accident. Express g as a function of p. Then evaluate g when p = 0.03. Also, find the value of p for which the company's expected overall profit is exactly 0.

Page 1 of 2

Problem 2 [40 marks]

A continuous random variable Y has the following probability density function (pdf):

f y( ) =?????c ece/?2?y,y2, yy = 00????? (? 0).

(a) Determine c as a function of ?. Then, for the case ?=2, evaluate c, calculate the maximum value of f y( ), and show this value in a sketch of f y( ).

(b) Determine F y( ) , the cumulative distribution function (cdf) of Y. Then, for the case ?= 2, calculate the value of F(0) and show this value in a sketch of F y( ) .

(c) Determine µ= EY as a function of ?, and evaluate µ for the case ?=2. Also determine the value of ? for which µ= 0 .

(d) Determine m = Median Y( ) as a function of ?, and then evaluate this function for ? = 1, 2, 3, 5 and 10, correct to at least one significant digit. Also, calculate ?0 , the value of ? for which m = 0. Then sketch m over the range 1= =? 10, and mark in the points at ? = 1, 2, 3, 5, 10 and ?0 .

Problem 3 [30 marks]

Suppose that X Y, ~ iid U(0,1). Derive and sketch the density of:

(a) R = Y / (1- X )

(b) R =Y / (Y - X ) .

In each case, calculate and mark in your sketch the mode (M), median (m), lower quartile (q) and upper quartile (Q) of R.

Note: The lower quartile of a random variable Z is defined as any value z such that:

P Z( = =z) 0.25, and P Z( = =z) 0.75.

Likewise, the upper quartile of Z is defined as any value z such that:

P Z( = =z) 0.75, and P Z( = =z) 0.25.

Page 2 of 2

AND STATISTICS

INTRODUCTORY MATHEMATICAL STATISTICS PRINCIPLES OF MATHEMATICAL STATISTICS

(STAT2001/6039)

Assignment 2 (2018 Semester 1)

Your solutions to this assignment should be placed in the appropriate box in the RSFAS School foyer by the due time and date (as provided in the Course Outline and on Wattle). Attach a cover sheet (as provided on Wattle) which has your ANU ID number. The assignment is out of 100 and is worth 10% of your overall course mark. Each problem has equally weighted parts. Show all relevant working and draw a box around each required result. Unless otherwise indicated, present final numerical solutions as exact simple fractions or correct to five significant digits (e.g. 0.024380).

Problem 1 [30 marks]

An insurance company offers a particular kind of policy whereby the insured person pays a premium of $200 and thereby becomes eligible to receive compensation in the event of a particular kind of accident whose probability of occurring during the policy period is 100p%. The policy will pay the costs arising from the accident, assumed to be exponential with mean $900, but only that part of the costs in excess of $500, and only up to a maximum payout of $4,000. Assume that at most one accident can occur.

(a) Find a, the probability that a policy holder will incur an overall loss over the policy period of more than $2,000, taking into account the premium and costs associated with the potential accident. Also, find ß, the probability that a person without the policy (but subject to the same risks as a policy holder) will incur an overall loss over the same period of more than $2,000, taking into account the costs associated with the potential accident. Express a and ß as functions of p. Then evaluate a and ß when p = 0.03.

(b) Find m, the expected value of the overall loss which a policy holder will incur over the policy period, taking into account the premium and the costs associated with the potential accident. Express m as a function of p. Then evaluate m when p = 0.03.

(c) Find g, the expected value of the overall profit which the insurance company will make from a single policy, taking into account the premium and the costs associated with the potential accident. Express g as a function of p. Then evaluate g when p = 0.03. Also, find the value of p for which the company's expected overall profit is exactly 0.

Page 1 of 2

Problem 2 [40 marks]

A continuous random variable Y has the following probability density function (pdf):

f y( ) =?????c ece/?2?y,y2, yy = 00????? (? 0).

(a) Determine c as a function of ?. Then, for the case ?=2, evaluate c, calculate the maximum value of f y( ), and show this value in a sketch of f y( ).

(b) Determine F y( ) , the cumulative distribution function (cdf) of Y. Then, for the case ?= 2, calculate the value of F(0) and show this value in a sketch of F y( ) .

(c) Determine µ= EY as a function of ?, and evaluate µ for the case ?=2. Also determine the value of ? for which µ= 0 .

(d) Determine m = Median Y( ) as a function of ?, and then evaluate this function for ? = 1, 2, 3, 5 and 10, correct to at least one significant digit. Also, calculate ?0 , the value of ? for which m = 0. Then sketch m over the range 1= =? 10, and mark in the points at ? = 1, 2, 3, 5, 10 and ?0 .

Problem 3 [30 marks]

Suppose that X Y, ~ iid U(0,1). Derive and sketch the density of:

(a) R = Y / (1- X )

(b) R =Y / (Y - X ) .

In each case, calculate and mark in your sketch the mode (M), median (m), lower quartile (q) and upper quartile (Q) of R.

Note: The lower quartile of a random variable Z is defined as any value z such that:

P Z( = =z) 0.25, and P Z( = =z) 0.75.

Likewise, the upper quartile of Z is defined as any value z such that:

P Z( = =z) 0.75, and P Z( = =z) 0.25.

Page 2 of 2