1.3. An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company’s employees that choose coverages A, B, and C are (1/4), (1/3), and (5/12), respectively. Determine the probability that a randomly chosen employee will choose no supplementary coverage.

A) 0                B) 47/144            C) ½          D) 97/144                    E) 7/9

C) 1/2

1.8. Under tan insurance policy, a maximum of five claims may be filed per year by a policyholder. Let p_n be the probability that a policyholder files n claims during a given year where n = 0, 1, 2, 3, 4, 5. An actuary makes the following observations:

        i.                 I)   P_n >= p_n+1 for n = 0, 1, 2, 3, 4.

      ii.                I I) The difference between p_n and p_n+1 is the same for n = 0, 1, 2, 3, 4.

    iii.              III)   Exactly 40% of policyholders file fewer than two claims during a           given year.

Calculate the probability that a random policyholder will file more than three claims during a given year.

A)    A)0.14                 B) 0.15            C) 0.27            D) 0.29            E) 0.33

C) 2.7

2.7. Let A, B and C be events such that P[A|C] =.05 and P[B|C]=.05. Which of the following statements must be true?

A) P[AՈB|C] =.05²

B) P[A’ՈB’|C] ≥ .90

C) P[AՍB|C] ≤ .05

D) P[AՍB|C’] ≥ 1- (.05)²

E) P[AՍB|C’] ≥ .10

B)

3.7. Three boxes are numbered 1, 2 and 3. For k = 1, 2, 3, box k contains k blue marbles and 5 – k red marbles. In a two-step experiment, a box is selected and 2 marbles are drawn from it without replacement. If the probability of selecting a box k is proportional to k, what is the probability that the two marbles drawn have different colors?

A) 17/60               B) 34/75          C) ½                D) 8/15            E) 17/30

E) 17/30

3.8. In Canada’s national 6-49 lottery, a ticket has 6 numbers each from 1 to 49, with no repeats. Find the probability of matching exactly 4 out of the 6 winning numbers if the winning numbers are all randomly chosen

A) .00095      B) .00097      C) .00099     D) .00101    E) .00103

B).00097

3.9. A number X is chosen at random from the series 2, 5, 8, … and another number Y is chosen at random from the series 3, 7, 11, … Each series has 100 terms. Find P[X = Y].

A) .0025          B) .0023          C) .0030                D) .0021          E) .0033 

A) .0025

3.11) A store has 80 modems in its inventory, 30 coming from Source A and the remainder from Source B. Of the modems from Source A, 20% are defective. Of the modems from Source B, 8% are defective. Calculate the probability that exactly two out of a random sample of five modems from the store’s inventory are defective.

 A. 0.010     B. 0.078    C. 0.102      D. 0.105      E. 0.125

C)0.102

4.13. For two random variables, the “distance” between two distributions is defined to be the maximum, max |F₁ (x) – F₂ (x)| over the range for which F1 and F2 are defined, where F(x) is the cumulative distribution function. Find the distance between the following two distributions:

        i.            Uniform on the interval [0, 1]

      ii.            Pdf is f(x) = 1/(x+1)^2 for 0 < x < ∞

A) 0                 B) ¼                C) ½                D) ¾                E) 1

C) 1/2

E) 38

C) 1200

C) .27

2. Let X be a continuous random variable with density function f(x) =  6x(1 − x), for 0 < x < 10,

0,     otherwise.

Find P[|X − 1/2|> 1/4 ].

A. .0521 B. .1563 C. .3125 D. .5000 E. .8000

A) 0.3125

In modeling the number of claims filed by an individual under an automobile policy during a three-year period, an actuary makes the simplifying assumption that for all integers n ≥ 0, P(n+1) = (1/5)*P(n) , where pn represents the probability that the policyholder files n claims during the period. Under this assumption, what is the probability that a policyholder files more than one claim during the period?

A. 0.04 B. 0.16 C. 0.20 D. 0.80 E. 0.96

c).04

A)0.2857

5.21.

A life insurer has created a special one-year term insurance policy for a pair of business people who travel to high-risk locations. The insurance policy pays nothing if neither die in the year, $100,000 if exactly one of the two die, and $K > 0 if both die. The insurer determines that there is a probability 0.1 that at least one of the two will die during the year and a probability of 0.08 that exactly one of the two will die during the year. You are told that the standard deviation of the payout is $74,000. Find the expected payout for the year on this policy.

A) 18,000  B) 21,000  C) 24,000  D) 27,000    E) 30,000

A) 1800

5.24)

Smith is offered the following gamble: he is to choose a coin at random from a large collection of coins and toss it randomly. 3/4 of the coins in the collection are loaded towards a head and 1/4 of the coains are loaded towards a tail. If a coin is loaded towards a head, then when the coin is tossed randomly, there is 3/4 probability that a head will turn up and a 1/4 probability that a tail will turn up. Similarly, if the coin is loaded towards tails, then there is a 3/4 probability that a tail will turn up and a 1/4 probability that a head will turn up. If Smith tosses a head, he loses $100, and if he tosses a tail, he wins $200. Smith is allowed to obtain “sample information” about the gamble. When he chooses the coin at random, he is allowed to toss it once before deciding to accept the gamble with that same coin. Suppose Smith tosses a head on the sample toss. Find Smith’s Expected gain/loss on the gamble if it is accepted.

A) loss of 20 B) loss of 10 C) loss of 0 D) gain of 10 E) gain of 20

C) loss of 10

5.4 

An insurer’s annual weather-related loss, X, is a random variable with density function

f (X)  = [2.5 * (200)^2.5] /[(x)^3.5], for x > 200,

0, otherwise.

Calculate the difference between the 30-th and 70-th percentiles of X.

A. 35 B. 93 C. 124 D. 131 E. 298

B)93

E) 374

B)50,400

C)5/36

5.20

An actuary uses the following distribution for the random variable T, time until death, for a new born baby: f(t) = t/5000 for 0 < t < 1000 . At the time of birth, an insurance policy is set up to pay an amount of (1.1)^t at time t if death occurs at that instant.Find the expected payout on this insurance policy. (nearest 100)

a) 2000 b) 2200 c) 2400 d) 2600 e) 2800

D)2600

An insurance company’s monthly claims are modeled by a continuous, positive random variable X, whose probability density function is proportional to

(1+ x )^-4 , where 0 < x < infinity .

Determine the company’s expected monthly claims.

A. 1/6 B. 1/3 C. 1/2 D. 1 E. 3

C)1/2

B)-.6

C)

A)

A).45

A system made up of 7 components with independent, identically distributed lifetimes will operate until any of 1 of the system's components fails. If the lifetime of
each component has density function ,

f(x) =3/(x^4)  for 1<x

 0,otherwise

 what is the expected lifetime until failure of the system? A) 1.02 B)1.03 C) 1.04 D) 1.05 E) 1.06 

D) 1.05

6.1)

X has a discrete uniform distribution on the integers 0,1,2,...n and Y has a discrete uniform distribution on the integers 1,2,3,...n.

Find Var[X] -Var[Y]

A)     (2n+1)/12 B) 1/12   )0   D) -1/12  E) –(2n+1)/12

A) (2n+1)/12

6.7)

Let X be a Poisson random variable with E[X] = ln 2. Calculate E[cos(Pi*X)].

A)     0          B)    ¼        C)    ½      D) 1     E) 2 ln 2  

B) 1/4

6.14)

A box contains 10 white marbles and 15 black marbles. Let X denote the number of white marbles in a selection of 10 marbles selected at random and without replacement. Find Var(x) / E(x)

A)     1/8   B)   3/16     C)  2/8      D) 5/16           E) 3/8

E) 3/8

6.21)

An insurer uses the Poisson distribution with mean 4 as the model for the number of warranty claims per month on a particular product. Each warranty claim results in a payment of 2 by the insurer. Find the probability that the total payment by the insurer in a given month is less than two standard deviations above the average monthly payment. 

A)     .9   B) .8    C) .7    D) .6   E) .5

B).8

6.23)

Let X be a random variable with moment generating function

 M(t)= [(2+e^t)/3]^9     ,   -infinity < t < infinity

Calculate the variance of X

A)     2    B) 3   C) 8    D) 9     E) 11

A) 2

6.27)

For a certain discrete random variable on the non-negative integers, the prob-
ability function satisfies the relationships
P(0) = P(1),
P(k + 1) = 1/k P(k),  for k = 1,2,3 ...

Find P(0)

A)     Ln e  B) e-1   C) (e+1)^-1   D) e^-1     E) (e-1)^-1

C) (e+1)^-1

6.5)

A hospital receives 1/5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. For Company X’s shipments, 10% of the vials are ineffective. For every other company, 2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X?

A. 0.10 B. 0.14 C. 0.37 D. 0.63 E. 0.86

A)0.10

6.6)

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 level during the coming year, independent of any other employee. Determine the maximum value of C for which the probability is less than 1% that the fund will be inadequate to cover all payments for high performance.

A. 24 B. 30 C. 40 D. 60 E. 120

D) 60

6.25)

A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is 3/5 . The number of accidents that occur in any given month is independent of the number of accidents that occur in all other months. Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs.

A. 0.01 B. 0.12 C. 0.23 D. 0.29 E. 0.41

D).29

6.26)

Each time a hurricane arrives, a new home has a 0.4 probability of experiencing damage. The occurrences of damage in different hurricanes are independent. Calculate the mode of the number of hurricanes it takes for the home to experience damage from two hurricanes.

A. 2 B. 3 C. 4 D. 5 E. 6 

B) 3

6.28)

Let X represent the number of customers arriving during the morning hours and let Y represent the number of customers arriving during the afternoon hours at a diner. You are given:

i) X and Y are Poisson distributed.

ii) The first moment of X is less than the first moment of Y by 8.

iii) The second moment of X is 60% of the second moment of Y. Calculate the variance of Y.

A. 4 B. 12 C. 16 D. 27 E. 35 

E) 35

A) .79

7.8)

For Company A there is a 60% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 10000 and standard deviation 2000. For Company B there is a 70% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 9000 and standard deviation 2000. Assume that the total claim amounts of the two companies are independent. What is the probability that, in the coming year Company B’s total claim amount will exceed Company A’s total claim amount?

A. 0.180 B. 0.185 C. 0.217 D. 0.223 E. 0.240 

D)0.223

10) Let X be a continuous random variable with density function     

f(x) = 1/sqrt(2pi)* e^[-(x^2)/2] for -infinity< x < infinity. Calculate E[X|X>0].

A)     0     B) 1/sqrt(2*Pi)    C) ½      D)   sqrt(2/*Pi)       E) 1

edit this card when you get answer

7.13 (SOA) The lifetime of a printer costing 200 is exponentially distributed with mean 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 100 printers, how much should it expect to pay in refunds?

A) 6,321     B) 7,358   C) 7,869    D) 10,256    E) 12,642

D) 10,256

7.14) (SOA) A piece of equipment is being insured against early failure. The time from purchase until failure of the equipment is exponentially distributed with mean 10 years. The insurance will pay an amount x if the equipment fails during the first year, and it will pay 0.5x if failure occurs during the second or third year. If failure occurs after the first three years, no payment will be made. At what level must x be set if the expected payment made under this insurance is to be 1000?

A) 3858    B) 4449  C) 5382     D) 5644      E) 7235

D) 5644

7.17) Average loss size per policy on a portfolio of policies is 100. Actuary 1 assumes that the distribution of loss size has an exponential distribution with a mean of 100, and Actuary 2 assumes that the distribution of loss size has a pdf of f2(x)=(2θ^2)/(x+θ)^3. If M1 and M2 represent the median loss sizes for the two distributions, find (M1)/(M2).

A) .6                      B)1.0                     C) 1.3                   D)1.7                    E)2.0

D)1.7

7.18) The time until the occurrence of a major hurricane is exponentially distributed. It is found that it is 1.5 times as likely that a major hurricane will occur in the next ten years as it is that the next major hurricane will occur in the next five years. Find the expected time until the next major hurricane.

A)5      B)5ln2      C) 5/(ln2)       D) 10ln2      E) 10/(ln2) 

C) 5/(ln2)

7.19) (SOA) A driver and a passenger are in a car accident. Each of the independently accidents has a probability of 0.3 of being hospitalized. When a hospitalization occurs, the loss is uniformly distributed on [0,1]. When two hospitalizations occur, the losses are independent. Calculate the expected number of people in the car who are hospitalized, given that the total loss due to hospitalization from the accident is less than 1.

A) 0.510    B) 0.534  C) 0.600    D) 0.628      E)0.800

B) 0.534