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### Homework Assignment 1

Course: STOR 472, Spring 2011
School: University of North...
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Word Count: 420

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Assignment Homework 1: (1) Let f(x) = a(e-2x e-3x) for x 0 and 0 elsewhere. Find the constant a so that f(x) satisfies the conditions of being a probability density function. Calculate Pr(X 1). (Ans a = 6; Pr(X 1) = 0.6936) Find the distribution function of X and calculate Pr ( X ). (Ans F(x) = 0 for x &lt; 0, .75x2 + .25x for 0 x 1 and 1 for x &gt; 1. Pr( X ) = 0.5) (2) Let f(x) = 1.5x +...

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Assignment Homework 1: (1) Let f(x) = a(e-2x e-3x) for x 0 and 0 elsewhere. Find the constant a so that f(x) satisfies the conditions of being a probability density function. Calculate Pr(X 1). (Ans a = 6; Pr(X 1) = 0.6936) Find the distribution function of X and calculate Pr ( X ). (Ans F(x) = 0 for x < 0, .75x2 + .25x for 0 x 1 and 1 for x > 1. Pr( X ) = 0.5) (2) Let f(x) = 1.5x + .25, 0 x 1, zero elsewhere, be the probability density function of X. (3) You play a slot machine repeatedly. The probability of winning on a single play is 0.05, and successive plays are independent. Let X be the random variable for the number of unsuccessful attempts before the first win. Find F(x) for x = 0,1,2, , and determine how many times you would need to play the slot machine in order to be sure that your probability of winning at least once is 99%. (Ans. F(x) = 1 - .95x+1, x = 0,1,2, . You need 90 plays for the probability to be 99% that you win at least once.) (4) Consider the following experiment: Start by tossing a fair coin. If it comes up heads, toss it again and take the result of the experiment be to a 1 if you get heads and a 0 if tails. But if the original toss comes up tails, then chose a number from the uniform distribution on (0,1) and let that be the result of the experiment. Find the distribution function of the outcome of this experiment and sketch its graph. [Ans F(x) = 0 if x < 0, F(x) = if x = 0, F(x) = (2x+1)/4 if 0<x<1, F(x) = 1 if x 1] Work (5a) if you have taken STOR 372; otherwise work (5b). (5a) Read through the solution to Example 4.2.4 on pages 103-105 in the textbook. Find the probability density function of Z. Are there any points at which f(z) is not defined? Use f(z) to find E(Z). You should get the same answer that is derived in part (a) of the solution. Ans: f(z) = 0 .1813 .4z-.6 for z = 0 for 0 < z < v5 for z < 0 and z > v5 it is undefined at z = v5 (5b) The distribution function of X is given by 1 1/x F(x) = 0 for x 1 for x < 1 Find the probability density function of X and sketch the graphs of both F(x) and f(x) . Are there any points at which f(x) is not defined? Ans: f(x) = 0 for x < 1 1/x2 for x > 1 it is undefined at x = 1
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University of North Carolina School of the Arts - STOR - 472
Homework Assignment 10: (1) An insurance company is selling the one-year term insurance that pays an extra benefit in case ofaccidental death that is described on page 29. The company wishes to have at least a 95% probability that premiums with a relativ
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 21:(1) Find a formula for (u), given that = 4/5 and p(x) is given by (1/9)(3e-3x) + (8/9)(6e-6x), x&gt;0. (The example I worked in class last week illustrates one solution. Another way to solve the problem is developed in section 13.6.)
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 2: (1) For a certain type of policy an insurance company divides its claims into two classes, minor andmajor. Last year 90% of the policyholders filed no claims, 9 percent filed minor claims and 1 percent filed major claims. The amoun
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 3:(1) A company establishes a fund of 120 from which it wants to pay an amount, C, to any of its 20 employees who achieve a high performance level during the coming year. Each employee has a 2% chance of achieving a high performance l
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 4: (1) An insurance policy is written to cover a loss X, where X has a uniform distribution on [0, 1000]. At what level must a deductible be set in order for the expected payment to be 25% of what it would be with no deductible? (Ans 5
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 5:(1) You are given: (i) Losses follow an exponential distribution with the same mean in all years. (ii) The loss elimination ratio this year is 70%. (iii) The deductible for the coming year is 4/3 of the current deductible. Compute t
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 6:(1) For 2003, loss sizes follow a uniform distribution with f(x) = 1 / 2500, 0 &lt; x &lt; 2500. Inflation of 3% impacts all losses uniformly from 2003 to 2004. In 2004 a deductible of 100 is applied to all losses. Determine the expected
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 7:(1) Let f(x) be the probability density function of the Pareto random variable; that is, f(x) = / (x+ ) + 1 , &gt; 0, &gt; 0,x &gt; 0. Show that F(x) = 1 [ / (x + )] and (x) = / (x + ). (2) Use the hazard rate function to show that the Weibu
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 8: (1) The probability of a fire in a certain structure in a given time is 0.05. If a fire occurs, the damage to the structure is uniformly distributed between 0 and 20. The claim random variable, X, represents the fire damage to the s
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 11:(1) An insurer offers group term insurance on 250 mutually independent lives for a premium of 285. The probability of a claim is 0.02 for each life. The distribution of the number of lives by benefit amount is: Benefit Amount Numbe
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 12: (1) In a clinic, physicians volunteer their time on a daily basis to provide care to those who are not eligible to obtain care otherwise. The number of physicians who volunteer in any day is uniformly distributed on the integers 1
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 13: (1) Prescription drug losses, S, are modeled assuming the number of claims has a geometric distribution with mean 4, and the amount of each prescription is 40. Calculate E[(S-100)+]. (Ans 92.16) (2) Do Exercise 12.10 on page 394. [
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 14: (1) Do Exercise 12.13 on page 394. (2) Lucky Lisa finds coins on her way to class at a Poisson rate of 0.5 coins per minute. The denominations are randomly distributed: (i) (ii) (iii) 60% of the coins are pennies; 20% of the coins
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 15:(1) The random variable S follows a binomial distribution with n = 3 and p = 1/3. The following procedure simulates S: Generate a uniform random number U on [0,1] and let 0, if U a 1, if a &lt; U b 2, if b &lt; U c 3, if c &lt; US=Determi
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 16:(1) You are the agent for a baseball player who desires an incentive contract that will pay the following amounts: Type of hit Probability of hit per time at bat Compensation per hit Single 0.14 x Double 0.05 2x Triple 0.02 3x Home
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 17:(1) For an allosaur with 10,000 calories stored at the start of a day: (i) The allosaur uses calories uniformly at a rate of 5,000 per day. If his stored calories reach 0, he dies. (ii) Each day the allosaur eats 1 actuary (10,000
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 18:For a claim number process cfw_N(t), t 0, you are given that the waiting times between successive claims are independent and identically distributed with distribution function F(t) = 1 e-2t. Determine the probability that exactly 3
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 19:(1) A continuous-time surplus process has a compound Poisson claims process. (i) The claim amount distribution is inverse Gaussian with = 1.0 and = 0.02. (ii) The relative security loading is positive. (iii) The adjustment coeffici
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 20:(1)(a) If the adjustment coefficient R = 2, find u so that (u) &lt; 0.05. (b) Suppose that X has a Gamma(2,3) distribution and that the adjustment coefficient R = 1. Find , the relative security loading. (c) Suppose that X is exponen
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 22:(1) Do Exercise 14.5 on page 462. (In part (b), the first method refers to the approximation we developed in class.) (2) Consider a group life insurance contract with an accidental death benefit. Assume that for all members the pro
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 23:(1) An insurance company sold one-year term life insurance on a group of 2,300 independent lives as given below: Class Benefit Amount Prob of Death Number of Policies 1 1 0.10 500 2 2 0.02 500 3 3 0.02 500 4 2 0.10 300 5 2 0.10 500
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 24:(1) Aggregate claims has a compound Poisson distribution with = 1 and p(1) = p(2) = 0.5. For a premium of 4.0, an insurer will pay total claims and a dividend equal to the excess, if any, of 75% of the premium over 100% of the clai
University of North Carolina School of the Arts - STOR - 472
Homework Assignment 25:(1) Using the table on page 449, calculate the expected value of the reinsurers payment for each of the following reinsurance arrangements: (a) The reinsurer pays 50% of aggregate claims in excess of 2. The maximum amount the reins
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