# Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

Download "Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions"

## Transcription

1 Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years, grouped by topic, and ordered by difficulty. All of the exams have appeared on the SOA website at some point in the past, though most of them are no longer there. The exams are copy-righted by the SOA. Note on the topics covered: This problem set covers problems on continuous random variables (Chapter 3 of Hogg/Tanis). Note on the ordering of the problems: The problems are loosely grouped by topic, and very roughly ordered by difficulty (though that ordering is very subjective). The last group of problems (those with triple digit numbers) are harder, either because they are conceptually more difficult, or simply because they require lengthy computations and may take two or three times as long to solve than an average problem. Answers and solutions: I will post an answer key and hints/solutions on the course webpage, hildebr/37.

2 . [3-] An insurance company issued insurance policies to 32 independent risks. For each policy, the probability of a claim is /6. The benefit given that there is a claim has probability density function 2( y), < y <, f(y) =, otherwise. Calculate the expected value of total benefits paid. (A) 6 9 (B) 8 3 (C) 32 9 (D) 6 3 (E) 32 3 Answer: A: 6/9 Hint/Solution: The expected value of a single benefit is E(Y ) = yf(y)dy = (2y 2y 2 )dy = /3. The expected number of claims is 32 (/6) = 6/3. Multiplying these two numbers gives the expected value of all benefits paid, (/3)(6/3) = 6/9. 2. [3-2] Let X be a continuous random variable with density function x for 2 x 4, Calculate the expected value of X. (A) 5 (B) 3 5 (C) (D) 28 5 (E) 2 5 Answer: D: 28/5 Hint/Solution: This is an easy integration exercise. The only tricky part here is the correct handling of the absolute value x appearing in the definition of f(x): One has to split the integral into two parts, corresponding to ranges 2 x and < x 4, replacing x by x in the first range, and by x in the second range, before evaluating the integral. 3. [3-3] A large company has determined that the function 3x 2, x, serves as the payroll density function. That is, the distribution of payroll, F (x), which is the proportion of total payroll earned by the lowest paid fraction x of employees, x, relates to f(x) in the same way that probability distributions and probability densities relate. Gini s index, G, defined as G = 2 x F (x) dx is a measure of how evenly payroll is distributed among all employees. Calculate G for this large company. Answer: C:.5 (A).2 (B).4 (C).5 (D).8 (E). Hint/Solution: This reduces to an easy integration problem. First compute F (x) by integrating f(x), then compute the integral in the definition of G. 2

3 4. [3-4] The loss amount, X, for a medical insurance policy has cumulative distribution function, ( ) x <, F (x) = 9 2x 2 x3 3, x 3,, x > 3. Calculate the mode of the distribution. Answer: D: 2 (A) 2 3 (B) (C) 3 2 (D) 2 (E) 3 Hint/Solution: The mode of a distribution is the point where f(x) is maximal. With the given distribution we have F (x) = (4/9)x (/9)x 2 for < x < 3, and outside the interval [, 3]. Differentiating, we get f (x) = (4/9) (2/9)x, and setting this equal to, we see that x = 2 is the only critical point of f. Hence the mode must occur at x = [3-5] An insurance company s monthly claims are modeled by a continuous, positive random variable X, whose probability density function is proportional to ( + x) 4, where < x <. Determine the company s expected monthly claims. Answer: C: /2 (A) 6 (B) 3 (C) 2 (D) (E) 3 Hint/Solution: We have c( + x) 4 for < x <, where c is a constant. First determine the proportionality constant c by setting c(+x) 4 equal to and solving for c. This gives c = 3. Then compute the integral E(X) = xf(x)dx = 3x( + x) 4 dx by substituting u = + x, du = dx. 6. [3-6] An insurance policy reimburses dental expense, X, up to a maximum benefit of 25. The probability density function for X is ce.4x for x, otherwise, where c is a constant. Calculate the median benefit for this policy. Answer: C: 73 (A) 6 (B) 65 (C) 73 (D) 82 (E) 25 Hint/Solution: The given distribution has the form of an exponential distribution, so the constant c must be.4, and the c.d.f. is equal to F (x) = e.4x. To obtain the median, set F (x) =.5 and solve for x: e.4x =.5, so.4x = ln.5, and x = ln(.5)/.4 = [3-7] 3

4 The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without failing for at least five hours. (A).7 (B).29 (C).38 (D).42 (E).57 Answer: D:.42 Hint/Solution: The general form of the c.d.f. of an exponential distribution is F (x) = e x/θ for x >. Since the median is 4 hours, we have F (4) = /2. This allows us to determine the parameter θ: e 4/θ = /2, so θ = 4/ ln 2. The probability that the component will work for at least 5 hours is given by F (5) = e 5/θ = e (5/4) ln 2 = [3-8] The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company expects that 3% of high-risk drivers will be involved in an accident during the first 5 days of a calendar year. What portion of high-risk drivers are expected to be involved in an accident during the first 8 days of a calendar year? (A).5 (B).34 (C).43 (D).57 (E).66 Answer: C:.43 Hint/Solution: If X denotes the number of days elapsed until an accident occurs, then we are given that X is exponentially distributed, and that P (X 5) =.3, and we have to compute P (X 8). By the general of the c.d.f. of an exponential distribution, we have F (x) = P (X x) = e x/θ for x >. By the given information, F (5) = P (X 5) =.3, which allows us to determine θ: We have e 5/θ =.3, so 5/θ = ln.7, or θ = 5/ ln.7. Substituting this back into F (x) and setting x = 8, we get F (8) = P (X 8) = e 8/θ = e (8/5) ln.7 = [3-9] The monthly profit of Company I can be modeled by a continuous random variable with density function f. Company II has a monthly profit that is twice that of Company I. Determine the probability density function of the monthly profit of Company II. Answer: (A) 2 f ( ) x 2 A (B) f ( ) x 2 (C) 2f ( ) x 2 (D) 2f(x) (E) 2f(2x) Hint/Solution: This is an easy exercise in changing variables in density functions if you apply to correct procedure for that. Let X denote the profit for Company I and Y that for Company II. Then Y = 2X. Let f(x) and F (x) denote the p.d.f. and c.d.f. of X, and let g(x) and G(x) denote the corresponding functions for Y. Then we have G(x) = P (Y x) = P (2X x) = P (X x/2) = F (x/2). Differentiating both sides with respect to x we get g(x) = G (x) = (/2)F (x/2) = (/2)f(x/2) since F (x) = f(x). Thus, the density of Y is (/2)f(x/2).. [3-] An insurance company insures a large number of homes. The insured value, X, of a randomly selected home is assumed to follow a distribution with density function 3x 4 for x >, 4

5 Given that a randomly selected home is insured for at least.5, what is the probability that it is insured for less than 2? (A).578 (B).684 (C).74 (D).829 (E).875 Answer: A:.578 Hint/Solution: Now, and We need to compute P (X 2 X.5) = P (X >.5) =.5 P (.5 X 2). P (X >.5) 3x 4 =.5 3 =.5 3 =.296, P (.5 X 2) = P (X >.5) P (X > 2) = =.7, so P (X 2 X.5) =.7/.296 = [3-] An insurance policy reimburses a loss up to a benefit limit of. The policyholder s loss, Y, follows a distribution with density function 2y 3 for y >, f(y) = What is the expected value of the benefit paid under the insurance policy? Answer: D:.9 (A). (B).3 (C).8 (D).9 (E) 2. Hint/Solution: The benefit, X, is given by X = Y if Y, and by X = if Y >. Thus, ( E(X) = y 2y 3 dy + 2y 3 dy = 2 ) + 2 =.9 2. [3-2] The lifetime of a machine part has a continuous distribution on the interval (, 4) with probability density function f, where f(x) is proportional to ( + x) 2. Calculate the probability that the lifetime of the machine part is less than 6. (A).4 (B).5 (C).47 (D).53 (E).94 Answer: C:.47 Hint/Solution: We are given that c( + x) 2 for x 4, with some constant c. To determine c, compute the integral of f(x) over the full interval (, 4) and set the integral equal to : = 4 c( + x) x dx = c((/) (/5)) = c.8, so c = /.8 = 2.5. The probability to compute is then equal to 6 c( + x) 2 dx = c((/) (/6)) = c(3/8) =.47. 5

6 3. [3-3] The time, T, that a manufacturing system is out of operation has cumulative distribution function ( ) 2 2 F (t) = t for t > 2, The resulting cost to the company is Y = T 2. Determine the density function of Y, for y > 4. Answer: (A) 4y 2 (B) 8y 3/2 (C) 8y 3 (D) 6y (E) 24y 5 A Hint/Solution: This is a routine application of the change of variables techniques; as always in these problems, you need to take a detour via the corresponding c.d.f. s to get the p.d.f. of the new variable. 4. [3-4] Under a group insurance policy, an insurer agrees to pay % of the medical bills incurred during the year by employees of a small company, up to a maximum total of one million dollars. The total amount of bills incurred, X, has probability density function x(4 x) 9 for < x < 3, otherwise, where x is measured in millions. Calculate the total amount, in millions of dollars, the insurer would expect to pay under this policy. (A).2 (B).3 (C).935 (D) (E) Answer: C:.935 Hint/Solution: Let Y denote the payout (in millions). Then Y = X if X, and Y = otherwise. Thus, 5. [3-5] E(Y ) = = = 9 xf(x)dx + f(x)dx 3 9 x2 (4 x)dx + x(4 x)dx 9 ( 4 3 ) + ( = =.935. The lifetime of a printer costing 2 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 printers, how much should it expect to pay in refunds? (A) 6, 32 (B) 7, 358 (C) 7, 869 (D), 256 (E) 2, 642 Answer: D:,256 ) 6

7 Hint/Solution: Let X denote the lifetime of the printer. Then the refund for each printer, Y, is given by Y = 2 if X, Y = if X 2, and Y = Thus, E(Y ) = 2P (X ) + P ( < X 2). The probabilities P (... ) here are easily computed using that fact X has exponential distribution with mean 2, and thus c.d.f. F (x) = P (X x) = e x/2 for x. Thus, P (X ) = e /2, P ( < X 2) = P (X 2) P (X ) = ( e 2/2 ) ( e /2 ) = e /2 e, and hence E(Y ) = 2 ( e /2 ) + (e /2 e ) = 2.5. Multiplying by the number of printers,, gives the total refund,, [3-6] An insurance policy is written to cover a loss, X, where X has a uniform distribution on [, ]. 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? Answer: C: 5 (A) 25 (B) 375 (C) 5 (D) 625 (E) 75 Hint/Solution: First note, that without the deductible the expected payment would be 5 (since it is then equal to the loss, and the loss X is uniformly distributed on [, ]). Next, let D denote the (unknown) deductible and Y the payment under deductible D. Then Y = X D if D X, and Y = if X D. Thus, E(Y ) = D (x D)f(x)dx = D (x D) dx = ( D)2 2 = ( D)2 2 Now set this expression equal to 25% of 5, i.e., 25, and solve for D: ( D) 2 /2 = 25, or ( D) 2 = 25, so D = [3-7] An insurance company sells an auto insurance policy that covers losses incurred by a policyholder, subject to a deductible of. Losses incurred follow an exponential distribution with mean 3. What is the 95th percentile of actual losses that exceed the deductible? (A) 6 (B) 7 (C) 8 (D) 9 (E) Answer: E: Hint/Solution: The main difficulty here is the correct interpretation of the 95th percentile of actual losses that exceed the deductible. The proper interpretation involves a conditional probability: we seek the value x such that the conditional probability that the loss is at most x, given that it exceeds the deductible, is.95, i.e., that P (X x X ) =.95, where X denotes the loss. By the complement formula for conditional probabilities, this is equivalent to P (X x X ) =.5. Since X is exponentially distributed with mean 3, we have P (X x) = e x/3, so for x >, P (X x X ) = P (X x) P (X = e x/3 e = /3 e (x )/3. Setting this equal to.5 and solving for x, we get (x )/3 = ln(.5), so x = 3 ln(.5) + =. 8. [3-8] 7

8 A piece of equipment is being insured against early failure. The time from purchase until failure of the equipment is exponentially distributed with mean years. The insurance will pay an amount x if the equipment fails during the first year, and it will pay.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? (A) 3858 (B) 4449 (C) 5382 (D) 5644 (E) 7235 Answer: D: 5644 Hint/Solution: Let T be the time until failure and X the payout. Then X = x if T, X = (/2)x if T 3, and X = (Note that x here is not the variable in a p.d.f. or c.d.f., but the unknown given in the problem that we need to determine.) Thus, E(X) = xp (T ) + (/2)xP ( T 3). Now P (T t) = e t/ by the given exponential distribution for T. Thus, P (T ) = e / =.96 and P ( T 3) = e / e 3/ =.63. Substituting these values into E(X) gives E(X) =.96x x =.77x. Setting this equal to and solving for x we get the answer, x = /.777 = [3-9] A manufacturer s annual losses follow a distribution with density function 2.5(.6) 2.5 x for x >.6, 3.5 To cover its losses, the manufacturer purchases an insurance policy with an annual deductible of 2. What is the mean of the manufacturer s annual losses not paid by the insurance policy? (A).84 (B).88 (C).93 (D).95 (E). Answer: C:.93 Hint/Solution: Let X denote the annual loss, and Y the part of the loss not covered by insurance. Then Y = X if X 2, and Y = 2 if X 2. To compute E(Y ), split the integral into intervals [.6, 2] and [2, ]. 2. [3-2] A device that continuously measures and records seismic activity is placed in a remote region. The time, T, to failure of this 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). Determine E(X). (A) e 6 (B) 2 2e 2/3 + 5e 4/3 (C) 3 (D) 2 + 3e 2/3 (E) 5 Answer: D 8

9 Hint/Solution: Since T is exponentially distributed with mean 3, the density of T is f(t) = (/3)e t/3 for t >. Since X = max(t, 2), we have X = 2 if T 2 and X = T if 2 < T <. Thus, 2. [3-2] E(X) = e t/3 dt + = 2( e 2/3 ) te t/3 + 2 t 3 e t/3 dt 2 2 e t/3 dt = 2( e 2/3 ) + 2e t/3 + 3e 2/3 = 2 + 3e 2/3 An insurance policy pays for a random loss X subject to a deductible of C, where < C <. The loss amount is modeled as a continuous random variable with density function 2x for < x <, Given a random loss X, the probability that the insurance payment is less than.5 is equal to.64. Calculate C. Answer: B:.3 (A). (B).3 (C).4 (D).6 (E).8 Hint/Solution: The insurance payment, Y, is given by Y = X C if X C, and Y = if X < C. Thus, Y.5 is equivalent to X.5 + C, and we have P (Y.5) = P (X C +.5) = C+.5 2xdx = (C +.5) 2. Setting this equal to.64 and solving for C we get (C +.5) 2 =.64, so C =.64.5 =.3. (Note that this argument would not be correct if C were greater than.5 since then the upper limit in the integral falls outside the range of the density. However, it is easy to see that this case cannot occur: Since the given density function restricts X to the interval X, if C > /2, then P (X.5+C) =, and hence also P (Y.5) =, contrary to the given condition P (Y.5) = [3-22] An actuary is reviewing a study she performed on the size of the claims made ten years ago under homeowners insurance policies. In her study, she concluded that the size of claims followed an exponential distribution and that the probability that a claim would be less than \$, was.25. The actuary feels that the conclusions she reached in her study are still valid today with one exception: every claim made today would be twice the size of a similar claim made ten years ago as a result of inflation. Calculate the probability that the size of a claim made today is less than \$,. Answer: C:.34 Hint/Solution: First determine the mean θ from the equation F () =.25, then compute F (5) (note that 5 is the claim size today corresponding to a claim size of ten years ago). 9

10 23. [3-23] A group insurance policy covers the medical claims of the employees of a small company. The value, V, of the claims made in one year is described by V =, Y, where Y is a random variable with density function k( y) 4 for < y <, f(y) = otherwise, where k is a constant. What is the conditional probability that V exceeds 4,, given that V exceeds,? (A).8 (B).3 (C).7 (D).2 (E).5 Answer: B:.3 Hint/Solution: First determine k by setting the integral over f(y) equal to : = f(y) = k ( y)4 dy = k(/5), so k = 5. The conditional probability to compute is P (Y 4, Y, ) = P (V.4 V.) = P (V.4) P (V.). Since V has density f(y) = 5( y) 4 for < y <, P (V.4) =.4 5( y)4 = (( ) 5 (.4) 5 ) =.6 5 and similarly P (V.) =.9 5, so the conditional probability sought is.6 5 /.9 5 = [3-24] The value, ν, of an appliance is given by ν(t) = e 7.2t, where t denotes the number of years since purchase. If appliance fails within seven years of purchase, a warranty pays the owner the value of the appliance at the time of failure. After seven years, the warranty pays nothing. The time until failure of the appliance has an exponential distribution with mean. Calculate the expected payment from the warranty. (A) 98.7 (B) 9.66 (C) (D) (E) Answer: D: Hint/Solution: Let X denote the payout under the warranty, and T the time until failure. Then X = e 7.2T if T 7, and X = Since T has exponential density with mean, its density function is (/)e t/ for t >. Thus, 25. [3-5] E(X) = 7 e 7.2t e t/ dt =. e 7 7 e.3t dt =. e 7 e = The loss due to a fire in a commercial building is modeled by a random variable X with density function.5(2 x) for < x < 2, Given that a fire loss exceeds 8, what is the probability that it exceeds 6?

11 Answer: B: /9 (A) /25 (B) /9 (C) /8 (D) /3 (E) 3/7 Hint/Solution: This is and easy integration exercise. We need to compute P (X 6 X 8), which is the same as P (X 6)/P (X 8). Integrating the given density function, we get P (X x) = 2 x.5(2 t)dt =.25(2 x) 2, < x < 2. Taking the ratio of these expressions with x = 6 and x = 8 gives the answer: (2 6) 2 /(2 8) 2 = / [3-52] An insurer s annual weather-related loss, X, is a random variable with density function 2.5(2) 2.5 x 3.5 for x > 2, Calculate the difference between the 3th and 7th percentiles of X. (A) 35 (B) 93 (C) 24 (D) 23 (E) 298 Answer: B: 93 Hint/Solution: We first compute the c.d.f. F (x) of X by integrating the given density: F (x) = x 2 2.5(2) 2.5 t 3.5 dt = ( ) 2.5 2, x > 2. x Setting F (x) equal to.3 and.7 and solving for x we get the 3th and 7th percentiles: x = 23.7 and x = The answer is the difference between these two numbers, [3-53] The warranty on a machine specifies that it will be replaced at failure or age 4, whichever occurs first. The machine s age at failure, X, has density function /5 for < x < 5, Let Y be the age of the machine at the time of replacement. Determine the variance of Y. (A).3 (B).4 (C).7 (D) 2. (E) 7.5 Answer: C:.7 Hint/Solution: We have Y = X if < X 5, 4 if 4 < X 5.

12 Thus, 28. [3-54] E(Y ) = E(Y 2 ) = 4 4 x 5 dt x 2 5 dt dt = 2.4, dt = Var(Y ) = E(Y 2 ) E(Y ) 2 =.77. The owner of an automobile insures it against damage by purchasing an insurance policy with a deductible of 25. In the event that the automobile is damaged, repair costs can be modeled by a uniform random variable on the interval (, 5). Determine the standard deviation of the insurance payment in the event that the automobile is damaged. Answer: B: 43 (A) 36 (B) 43 (C) 433 (D) 464 (E) 52 Hint/Solution: Let X denote the repair costs, and let Y denote the insurance payment. We need to compute σ = Var(Y ). We are given that X is uniformly distributed on the interval (, 5), so the density of X is /5 for < x < 5. Taking into account the 25 deductible, we see that Y is given by X 25 if 25 X 5, Y = Hence, 29. [3-55] E(Y ) = E(Y 2 ) = (x 25) 5 dx = , (x 25) dx = 5 3 5, σ = E(Y 2 ) E(Y ) 2 = An actuary models the lifetime of a device using the random variable Y = X.8, where X is an exponential random variable with mean year. Determine the probability density function f(y), for y >, of the random variable Y. (A) y.8 e 8y.2 (B) 8y.2 e y.8 (C) 8y.2 e.y.25 (D) (.y).25 e.25(.y).25 (E).25(.y).25 e (.y).25 Answer: E 2

13 Hint/Solution: This is a change of variables exercise. Let g(x) and G(x) denote the density and c.d.f. of X, and let f(y) and F (y) denote the density and c.d.f. of Y. Then, for y >, Differentiating with respect to y we get F (y) = P (Y y) = P (X.8 y) = P (X (.y).25 ) = G((.y).25 ). f(y) = F (y) = G ((.y).25 ).25(.y).25. = g((.y).25 ).25(.y).25 = e (.y).25.25(.y).25. since g(x) = e x by the exponential distribution of X with mean. 3. [3-56] A tour operator has a bus that can accommodate 2 tourists. The operator knows that tourists may not show up, so he sells 2 tickets. The probability that an individual tourist will not show up is.2, independent of all other tourists. Each ticket costs 5, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay (ticket cost + 5 penalty) to the tourist. What is the expected revenue of the tour operator? (A) 935 (B) 95 (C) 967 (D) 976 (E) 985 Answer: E:985 Hint/Solution: consider: This is simpler than it might seem since there are only two cases to (i) All 2 ticket holders show up. This occurs with probability (.2) 2 =.654. In this case, the operator earns 2 5 = 5 from ticket sales, but incurs a cost of for the one ticket holder that cannot be accommodated, so the revenue is 5 = 95. (ii) Not all 2 ticket holders show up. This occurs with the complementary probability,.654 =.346. In this case, the operator earns 5 from ticket sales, but incurs no penalty, so the revenue is 5. Hence the expected revenue is = [3-57] An investment account earns an annual interest rate R that follows a uniform distribution on the interval (.4,.8). The value of a, initial investment in this account after one year is given by V =, e R. Determine the cumulative distribution function, F (v), of V for values v that satisfy < F (v) <. (A), ev/,, (B) 25e v/,.4 v, 48 (C), 833, 48 (D) 25 v 3

14 ( ( ) ) v (E) 25 ln.4, Answer: E Hint/Solution: This is a change of variables exercise. Let g(r) and G(r) denote the density and c.d.f. of R, and let f(v) and F (v) denote the density and c.d.f. of Y. Then, for v in the appropriate range, F (v) = P (V v) = P (e R v) ( ( v )) ( ( v )) = P R ln = G ln. Since R has uniform distribution on the interval (.4,.8), G(r) is given by (for r in this interval) G(r) = r.4 = 25(r.4)..8.4 Substituting this above gives the correct answer, (E): ( ( ) ) v F (v) = 25 ln.4., 32. [3-6] You are given the following information about N, the annual number of claims for a randomly selected insured: P (N = ) = 2, P (N = ) = 3, P (N > ) = 6. Let S denote the total annual claim amount for an insured. When N =, S is exponentially distributed with mean 5. When N >, S is exponentially distributed with mean 8. Determine P (4 < S < 8). Answer: (A).4 (B).8 (C).2 (D).24 (E).25 C Hint/Solution: First note that, if S were a pure exponential variable with a fixed mean θ and c.d.f. F (x), then P (4 < S < 8) = F (8) F (4) = ( e 8/θ ) ( e 4/θ ) = e 4/θ e 8/θ. For the particular means θ = 5 and θ = 8, these probabilities become.823 and.397. The probability sought in the problem is a weighted average of these two probabilities, the weigths being the probabilities with which these two cases occur: /3 for mean 5 and /6 for mean 8. The result is.823(/3) +.397(/6) = [3-2] Let T denote the time in minutes for a customer service representative to respond to telephone inquiries. T is uniformly distributed on the interval with endpoints 8 minutes and 2 minutes. Let R denote the average rate, in customers per minute, at which the representative responds to inquiries. Which of the following is the density function of the random variable R on the interval (/2 r /8)? (A) 2 5 4

15 (B) 3 5 2r (C) 3r 5 ln r 2 (D) r 2 (E) 5 2r 2 Answer: E Hint/Solution: The key observation here is that R and T are related by R = /T. Since we know the distribution of T (uniform distribution on [8, 2]), the problem then becomes a change of variables problem for density functions. As usual, this requires a detour through the corresponding c.d.f. s. Let f(t) and F (t) denote the p.d.f. and c.d.f. of T, and let g(r) and G(r) denote the corresponding functions for R. By the given uniform distribution of T on [8, 2], we have f(t) = /4 for 8 t 2. Hence, for /2 r /8, ( ) ( G(r) = P (R r) = P T r = P T ) ( ) = F. r r Differentiating, we get 34. [3-3] g(r) = G (r) = F ( r ) ( r 2 ) = f ( ) r 2 = r 4 r 2 = 5 2r 2. An auto insurance company insures an automobile worth 5, for one year under a policy with a, deductible. During the policy year there is a.4 chance of partial damage to the car and a.2 chance of a total loss of the car. If there is partial damage to the car, the amount X of damage (in thousands) follows a distribution with density function.53e x/2 for < x < 5, What is the expected claim payment? Answer: B: 328 Hint/Solution: (A) 32 (B) 328 (C) 352 (D) 38 (E) 54 This is a very lengthy computation. Let Y be the claim payment in thousands, and let D and T stand for partial damage and total loss, respectively. We need to compute E(Y ). To this end we need to consider separately the case when there there is a partial loss and the case when there is a total loss, and compute the conditional expectation of Y under these case cases, E(Y D) and E(Y T ). The overall expectation then is given by E(Y ) = E(Y D)P (D) + E(Y T )P (T ). We are given that P (D) =.4 and P (T ) =.2. Moreover, in the case of a total loss, the payment is (in thousands) 5 = 4, so E(Y T ) = 4. Thus, it remains to compute E(Y D), the expected loss in case in case of partial damage. From the problem, we have in this case X if X < 5, Y = 5

16 Thus, 5 5 E(Y D) = (x )f(x)dx = (x ).53e x/2 dx = [( 2)(x ).53e x/2] e x/2 dx = ( 2)(4).53e 5/ (e /2 e 5/2) =.249 Hence, E(Y ) = =

### Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 1 Solutions

Math 70, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

### Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions)

Math 370, Actuarial Problemsolving Spring 008 A.J. Hildebrand Practice Test, 1/8/008 (with solutions) About this test. This is a practice test made up of a random collection of 0 problems from past Course

### Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2 Solutions

Math 370, Spring 008 Prof. A.J. Hildebrand Practice Test Solutions About this test. This is a practice test made up of a random collection of 5 problems from past Course /P actuarial exams. Most of the

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 2 Solutions

Math 70/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 2 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,

### Math 370, Spring 2008 Prof. A.J. Hildebrand. Practice Test 2

Math 370, Spring 2008 Prof. A.J. Hildebrand Practice Test 2 About this test. This is a practice test made up of a random collection of 15 problems from past Course 1/P actuarial exams. Most of the problems

### UNIVERSITY of TORONTO. Faculty of Arts and Science

UNIVERSITY of TORONTO Faculty of Arts and Science AUGUST 2005 EXAMINATION AT245HS uration - 3 hours Examination Aids: Non-programmable or SOA-approved calculator. Instruction:. There are 27 equally weighted

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 5 Solutions

Math 370/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 5 Solutions About this problem set: These are problems from Course 1/P actuarial exams that I have collected over the

### 1. A survey of a group s viewing habits over the last year revealed the following

1. A survey of a group s viewing habits over the last year revealed the following information: (i) 8% watched gymnastics (ii) 9% watched baseball (iii) 19% watched soccer (iv) 14% watched gymnastics and

### SOCIETY OF ACTUARIES EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS

SOCIETY OF ACTUARIES EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS Copyright 015 by the Society of Actuaries Some of the questions in this study note are taken from past examinations. Some of the questions

### SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS Copyright 005 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions in this study

### SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS Copyright 007 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions in this study

### Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Problem Set 1 (with solutions)

Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand Problem Set 1 (with solutions) About this problem set: These are problems from Course 1/P actuarial exams that I have collected over the years,

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 1

Math 370/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 1 About this problem set: These are problems from Course 1/P actuarial exams that I have collected over the years, grouped

### Continuous Probability Distributions (120 Points)

Economics : Statistics for Economics Problem Set 5 - Suggested Solutions University of Notre Dame Instructor: Julio Garín Spring 22 Continuous Probability Distributions 2 Points. A management consulting

### SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS

SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS Copyright 5 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions in this study

### STA 256: Statistics and Probability I

Al Nosedal. University of Toronto. Fall 2014 1 2 3 4 5 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Experiment, outcome, sample space, and

### Chapter 4 Expected Values

Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges

### Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

### 5. Continuous Random Variables

5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be

### 3. The Economics of Insurance

3. The Economics of Insurance Insurance is designed to protect against serious financial reversals that result from random evens intruding on the plan of individuals. Limitations on Insurance Protection

### Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density

HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,

### EDUCATION AND EXAMINATION COMMITTEE SOCIETY OF ACTUARIES RISK AND INSURANCE. Copyright 2005 by the Society of Actuaries

EDUCATION AND EXAMINATION COMMITTEE OF THE SOCIET OF ACTUARIES RISK AND INSURANCE by Judy Feldman Anderson, FSA and Robert L. Brown, FSA Copyright 25 by the Society of Actuaries The Education and Examination

### Contents. 3 Survival Distributions: Force of Mortality 37 Exercises Solutions... 51

Contents 1 Probability Review 1 1.1 Functions and moments...................................... 1 1.2 Probability distributions...................................... 2 1.2.1 Bernoulli distribution...................................

### November 2012 Course MLC Examination, Problem No. 1 For two lives, (80) and (90), with independent future lifetimes, you are given: k p 80+k

Solutions to the November 202 Course MLC Examination by Krzysztof Ostaszewski, http://www.krzysio.net, krzysio@krzysio.net Copyright 202 by Krzysztof Ostaszewski All rights reserved. No reproduction in

### ( ) is proportional to ( 10 + x)!2. Calculate the

PRACTICE EXAMINATION NUMBER 6. An insurance company eamines its pool of auto insurance customers and gathers the following information: i) All customers insure at least one car. ii) 64 of the customers

### Solution. Let us write s for the policy year. Then the mortality rate during year s is q 30+s 1. q 30+s 1

Solutions to the May 213 Course MLC Examination by Krzysztof Ostaszewski, http://wwwkrzysionet, krzysio@krzysionet Copyright 213 by Krzysztof Ostaszewski All rights reserved No reproduction in any form

### Math 461 Fall 2006 Test 2 Solutions

Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two

### COURSE 1 REVISED SAMPLE EXAM

COURSE REVISED SAMPLE EXAM A table of values for the normal distribution will be provided with the Course Exam. Revised August 999 Problem # A marketing survey indicates that 60% of the population owns

### University of California, Berkeley, Statistics 134: Concepts of Probability

University of California, Berkeley, Statistics 134: Concepts of Probability Michael Lugo, Spring 211 Exam 2 solutions 1. A fair twenty-sided die has its faces labeled 1, 2, 3,..., 2. The die is rolled

### Continuous random variables

Continuous random variables So far we have been concentrating on discrete random variables, whose distributions are not continuous. Now we deal with the so-called continuous random variables. A random

### ECE302 Spring 2006 HW3 Solutions February 2, 2006 1

ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### f(x) = lim 2) = 2 2 = 0 (c) Provide a rough sketch of f(x). Be sure to include your scale, intercepts and label your axis.

Math 16 - Final Exam Solutions - Fall 211 - Jaimos F Skriletz 1 Answer each of the following questions to the best of your ability. To receive full credit, answers must be supported by a sufficient amount

### ( ) = 1 x. ! 2x = 2. The region where that joint density is positive is indicated with dotted lines in the graph below. y = x

Errata for the ASM Study Manual for Exam P, Eleventh Edition By Dr. Krzysztof M. Ostaszewski, FSA, CERA, FSAS, CFA, MAAA Web site: http://www.krzysio.net E-mail: krzysio@krzysio.net Posted September 21,

### Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions

Chapter 4 - Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the

### Jointly Distributed Random Variables

Jointly Distributed Random Variables COMP 245 STATISTICS Dr N A Heard Contents 1 Jointly Distributed Random Variables 1 1.1 Definition......................................... 1 1.2 Joint cdfs..........................................

### assureflex health spending account the evolution of health and dental care benefits

assureflex health spending account the evolution of health and dental care benefits Savings Comparisons Assureflex Corporation Post Office Box 81, Strathroy, Ontario N7G 3J1 telephone: (519) 245-3283 local

### Practice Exam 1. x l x d x 50 1000 20 51 52 35 53 37

Practice Eam. You are given: (i) The following life table. (ii) 2q 52.758. l d 5 2 5 52 35 53 37 Determine d 5. (A) 2 (B) 2 (C) 22 (D) 24 (E) 26 2. For a Continuing Care Retirement Community, you are given

### Important Probability Distributions OPRE 6301

Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.

### Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

### Mathematics of Risk. Introduction. Case Study #1 Personal Auto Insurance Pricing. Mathematical Concepts Illustrated. Background

Mathematics of Risk Introduction There are many mechanisms that individuals and organizations use to protect themselves against the risk of financial loss. Government organizations and public and private

### Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

### **BEGINNING OF EXAMINATION**

November 00 Course 3 Society of Actuaries **BEGINNING OF EXAMINATION**. You are given: R = S T µ x 0. 04, 0 < x < 40 0. 05, x > 40 Calculate e o 5: 5. (A) 4.0 (B) 4.4 (C) 4.8 (D) 5. (E) 5.6 Course 3: November

### CHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.

Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,

### Particular Solutions. y = Ae 4x and y = 3 at x = 0 3 = Ae 4 0 3 = A y = 3e 4x

Particular Solutions If the differential equation is actually modeling something (like the cost of milk as a function of time) it is likely that you will know a specific value (like the fact that milk

### Example: 1. You have observed that the number of hits to your web site follow a Poisson distribution at a rate of 2 per day.

16 The Exponential Distribution Example: 1. You have observed that the number of hits to your web site follow a Poisson distribution at a rate of 2 per day. Let T be the time (in days) between hits. 2.

### STAT 3502. x 0 < x < 1

Solution - Assignment # STAT 350 Total mark=100 1. A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous

### ECE302 Spring 2006 HW5 Solutions February 21, 2006 1

ECE3 Spring 6 HW5 Solutions February 1, 6 1 Solutions to HW5 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics

### SOCIETY OF ACTUARIES. EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE QUESTIONS

SOCIETY OF ACTUARIES EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE QUESTIONS The following questions or solutions have been modified since this document was prepared to use with the syllabus effective

### Solutions to Study Guide for Test 3. Part 1 No Study Guide, No Calculator

Solutions to Study Guide for Test 3 Part 1 No Study Guide, No Calculator 1. State the definition of the derivative of a function. Solution: The derivative of a function f with respect to x is the function

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by

MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x

### Random Variables. Chapter 2. Random Variables 1

Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets

### Change of Continuous Random Variable

Change of Continuous Random Variable All you are responsible for from this lecture is how to implement the Engineer s Way (see page 4) to compute how the probability density function changes when we make

### 1. Let A, B and C are three events such that P(A) = 0.45, P(B) = 0.30, P(C) = 0.35,

1. Let A, B and C are three events such that PA =.4, PB =.3, PC =.3, P A B =.6, P A C =.6, P B C =., P A B C =.7. a Compute P A B, P A C, P B C. b Compute P A B C. c Compute the probability that exactly

### 7 CONTINUOUS PROBABILITY DISTRIBUTIONS

7 CONTINUOUS PROBABILITY DISTRIBUTIONS Chapter 7 Continuous Probability Distributions Objectives After studying this chapter you should understand the use of continuous probability distributions and the

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### Manual for SOA Exam MLC.

Chapter 4. Life Insurance. Extract from: Arcones Manual for the SOA Exam MLC. Fall 2009 Edition. available at http://www.actexmadriver.com/ 1/14 Level benefit insurance in the continuous case In this chapter,

sheng@mail.ncyu.edu.tw 1 Content Introduction Expectation and variance of continuous random variables Normal random variables Exponential random variables Other continuous distributions The distribution

### A Pricing Model for Underinsured Motorist Coverage

A Pricing Model for Underinsured Motorist Coverage by Matthew Buchalter ABSTRACT Underinsured Motorist (UIM) coverage, also known as Family Protection coverage, is a component of most Canadian personal

### 3. Continuous Random Variables

3. Continuous Random Variables A continuous random variable is one which can take any value in an interval (or union of intervals) The values that can be taken by such a variable cannot be listed. Such

### Statistics 100A Homework 3 Solutions

Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win \$ for each black ball selected and we

### 2 Integrating Both Sides

2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation

### The graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0

. Compute the area between the curves x y 4y and x y y. Let f(y) y 4y y(y 4). f(y) when y or y 4. Let g(y) y y y( y). g(y) when y or y. x 3 y? The graphs of f and g intersect at (, ) and one other point.

### WEEK #22: PDFs and CDFs, Measures of Center and Spread

WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook

### May 2012 Course MLC Examination, Problem No. 1 For a 2-year select and ultimate mortality model, you are given:

Solutions to the May 2012 Course MLC Examination by Krzysztof Ostaszewski, http://www.krzysio.net, krzysio@krzysio.net Copyright 2012 by Krzysztof Ostaszewski All rights reserved. No reproduction in any

### UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

TABLE OF CONTENTS 1. Survival A. Time of Death for a Person Aged x 1 B. Force of Mortality 7 C. Life Tables and the Deterministic Survivorship Group 19 D. Life Table Characteristics: Expectation of Life

### II. Sketch the given region R and then find the area. 2. R is the region bounded by the curves y = 0, y = x 2 and x = 3.

Math 34 April I. It is estimated that t days from now a farmer s crop will be increasing at the rate of.5t +.4t + bushels per day. By how much will the value of the crop increase during the next 5 days

### 4. Life Insurance. 4.1 Survival Distribution And Life Tables. Introduction. X, Age-at-death. T (x), time-until-death

4. Life Insurance 4.1 Survival Distribution And Life Tables Introduction X, Age-at-death T (x), time-until-death Life Table Engineers use life tables to study the reliability of complex mechanical and

### Continuous Random Variables

Chapter 5 Continuous Random Variables 5.1 Continuous Random Variables 1 5.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize and understand continuous

### Calculus 1: Sample Questions, Final Exam, Solutions

Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.

### The Basics of Interest Theory

Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### P(a X b) = f X (x)dx. A p.d.f. must integrate to one: f X (x)dx = 1. Z b

Continuous Random Variables The probability that a continuous random variable, X, has a value between a and b is computed by integrating its probability density function (p.d.f.) over the interval [a,b]:

### The random variable X - the no. of defective items when three electronic components are tested would be

RANDOM VARIABLES and PROBABILITY DISTRIBUTIONS Example: Give the sample space giving a detailed description of each possible outcome when three electronic components are tested, where N - denotes nondefective

### 14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style

Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of

### **BEGINNING OF EXAMINATION** The annual number of claims for an insured has probability function: , 0 < q < 1.

**BEGINNING OF EXAMINATION** 1. You are given: (i) The annual number of claims for an insured has probability function: 3 p x q q x x ( ) = ( 1 ) 3 x, x = 0,1,, 3 (ii) The prior density is π ( q) = q,

### Math 130 Winter 2010 Quotient/Product/Chain Rule Practice January 28, 2010

Math Winter 0 Quotient/Product/Chain Rule Practice January 8, 0 1. Find the derivative of the following functions: a) gx) = x + 4x) 5 + x x + 1. To do this problem we treat each part seperately. F the

### AP CALCULUS AB 2009 SCORING GUIDELINES

AP CALCULUS AB 2009 SCORING GUIDELINES Question 3 Mighty Cable Company manufactures cable that sells for \$120 per meter. For a cable of fixed length, the cost of producing a portion of the cable varies

### Solution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute

Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the

### Continuous Random Variables

Probability 2 - Notes 7 Continuous Random Variables Definition. A random variable X is said to be a continuous random variable if there is a function f X (x) (the probability density function or p.d.f.)

### MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables

MATH 10: Elementary Statistics and Probability Chapter 5: Continuous Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides,

### Probabilities and Random Variables

Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments so

### Name: ID: Discussion Section:

Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS:

### MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

### Section 5.1 Continuous Random Variables: Introduction

Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,

### Math 425 (Fall 08) Solutions Midterm 2 November 6, 2008

Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the

### We hope you will find this sample a useful supplement to your existing educational materials, and we look forward to receiving your comments.

To Teachers of Mathematics: Thank you for visiting the BeAnActuary.org booth at the annual meeting of the National Council of Teachers of Mathematics. BeAnActuary.org is sponsored by the Joint Career Encouragement

### Student Number: SOLUTION Page 1 of 14

Student Number: SOLUTION Page 1 of 14 QUEEN S UNIVERSITY FACULTY OF ARTS AND SCIENCE DEPARTMENT OF MATHEMATICS AND STATISTICS MATH126 December Examination December 14th, 2009 Instructors: P. Li (A), A.

### Recitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere

Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x

### Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x

### correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

### Manual for SOA Exam MLC.

Chapter 4. Life insurance. Extract from: Arcones Fall 2009 Edition, available at http://www.actexmadriver.com/ (#1, Exam M, Fall 2005) For a special whole life insurance on (x), you are given: (i) Z is

### Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page

Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8

### ST 371 (VIII): Theory of Joint Distributions

ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or

### Heriot-Watt University. BSc in Actuarial Mathematics and Statistics. Life Insurance Mathematics I. Extra Problems: Multiple Choice

Heriot-Watt University BSc in Actuarial Mathematics and Statistics Life Insurance Mathematics I Extra Problems: Multiple Choice These problems have been taken from Faculty and Institute of Actuaries exams.

### Probability density function : An arbitrary continuous random variable X is similarly described by its probability density function f x = f X

Week 6 notes : Continuous random variables and their probability densities WEEK 6 page 1 uniform, normal, gamma, exponential,chi-squared distributions, normal approx'n to the binomial Uniform [,1] random