Questions and Answers of Introduction To Actuarial And Financial

Use the Newton-Raphson method to determine all real roots of the function f(x) = e(x−1)2 − 2. An error tolerance of ϵ = 0.0001 should be used.
Use the Newton-Raphson method with error tolerance ϵ = 0.0001 to find the single real root of h(x) = ln(x) − 2 from the initial estimate x0 = 8.
Evaluate the integrals in Example 12.16.Example 12.16Using the form of Eq. (12.10), express the integral of f (x, y) = xy over the domains illustrated in Figure 12.10.
State the dimensions of the following matrices in m × n format. In each case, state any three separate entries using “aij” notation.
What is the probability that the second card drawn from a standard pack of playing cards is a king?
Determine the following using the complex numbers defined in Question 8.1.Question 8.1Classify the following as real, imaginary, or complex. Plot each zi on a single Argand diagram.a. z1 = 3b. z2 = 2
Use De Moivre’s formula to derive triple-angle identities for sin and cos functions.
Determine all values of the cube root of 1 in the complex plane.
Use integration by parts to compute the following definite integrals. Note that the equivalent indefinite integrals were considered in Example 6.9.Example 6.9Use integration by parts to compute the
Find the real roots of the logarithmic functions h(x) and l(x) defined in Eqs. (2.18) and (2.19).
Using the form of Eq. (12.10), express the integral of f (x, y) = xy over the domains illustrated in Figure 12.10. V = ff f(x, y) dx dy D (12.10)
Twenty otherwise identical balls are individually numbered 1-20 and placed inside a bag. Calculate the probabilities of the following events.a. A random draw of a single ball produces a ball numbered
Calculate the probability of the following outcomes from a single roll of a fair dice. a. b. Either or C. Any one of O, O, J, 8, 8, or
Consider a bag containing a number of balls colored either red, blue, green, or yellow, denotedrespectively. In particular, there areCalculate the probability that the draw of a single ball will be
A production line in a factory produces 50 units per week and, on average, 10% of these units are expected to have some defect. Calculate the probability that a random selection of two units will
A bag contains five balls colored redgreenblueyellowand blackState which of the following are valid partitions of the sample space. R
Calculate the probability of obtaining the following total scores from simultaneously rolling two fair dice.a. 1b. 12c. 6d. 7
The math department of a university has been moved temporarily to another building while their building is being modernized. Each professor is to be allocated an office at random. If the probability
Using the bag of balls in Example 9.3, calculate the probability of obtaining the following outcomes from two draws. State whether the draws are independent or dependent. a. b. 1st: G) is drawn then
Calculate the number of possible permutations and combinations of the following.a. Arranging three balls from a total of three distinct balls.b. Arranging three balls from a total of four distinct
A new medical test has recently been approved to test for a rare genetic condition. The test is known to be 99% accurate. You ask that your doctor perform this test on you and the result comes back
Consider a jar of 20 distinct marbles. How many ways are there are arranging a random draw of three marbles ifa. The order does not matter?b. The order does matter?
Determine how common the condition in Question 9.5 must be for you to be 90% sure that a positive test result means that you have the condition. You should still assume that the test is 99%
A single dice is rolled repeatedly until it results in Calculate theprobability that this occurs on roll n, where n is any even number, in each of the following situations. # #
Consider a bag containing 10 balls numberedBalls are drawn at random without replacement.a. Calculate the probability that a random selection of three balls isin any order.b. Calculate the
A winning lottery ticket is one that correctly matches n balls drawn in any order from the total of 49 distinct balls.a. Calculate the probability that a single lottery ticket will win if n = 6.b.
State the sample space associated with the following experiments.a. Tossing a coin.b. Drawing a single ball from a bag containing ballsc. Drawing two balls without replacement from the bag in part b.
Express the following in standard actuarial notation.a. The probability that a life aged 50 exact will survive to age 56 exact.b. The probability that a life aged 40 exact will die aged either 90,
The 1-year survival probability of a particular species of animal is known to be given bywhere x = 0, 1, 2, . . . is age.a. Calculate the probability that an animal aged 5 exact will survive another
Use set notation to express the following events associated with rolling a single dice.a. A is the event that we obtain b. B is the event that we obtain an even score.c. C is the event that we
Use set notation and Eqs. (9.3)–(9.6) to prove Eq. (9.8). P(A) = 1 P(A) (9.8)
Obtain and interpret the events formed from the following operations on the events defined in Example 9.10.a. Bcb. C ∪ Dc. A ∩ Bd. B ∩ CcExample 9.10Use set notation to express the following
A young man receives a new car and an unusual financial contract as a gift for his 16th birthday. If the car is worth $11,000 and the contract promises to pay $10,000 if he survives to his 20th
A bag containsballs. If a single ball is drawn at random from the bag, calculate the probability of each of the follow events. 1 x (Y), 3 x (B), 2x (G) G), and 4 x (R
Demonstrate Eqs. (9.8)–(9.10) using examples from the roll of a single dice. P(A) = 1 P(A) if A C B then P(A) ≤ P(B) P(AUB) = P(A) + P(B)- P(ANB) (9.8) (9.9) (9.10)
A bag contains four colored ballsBalls are drawn at random and replaced before the next draw. Calculate the probabilities of the following events. R G B), and (Y)
A single card is drawn from a standard deck of 52 playing cards. Calculate the probability of each of the following events.a. A1 is the event that the card is either a king or a heart.b. A2 is the
Two fair dice, one white and one black, are rolled. Calculate the probability of the following events. a. Aa = {80} b. Ab = {} c. Ac = {(80, 8}
Determine why the following are not valid partitions of the sample space of rolling a standard white dice. a. B" = {0,0} {0,0} b. B" = {0, c. B = B B = {0, (0,0,0 3, 0} = {0,0} B = {8, 8, 0) B" = {0,
Consider two bags of balls. Bag 1 containsand bag 2 containsWe now roll a fair dice. If the dice shows an even score we draw from bag 1, otherwise we draw from bag 2. Calculate the probability of
Three cards are drawn from a standard pack of playing cards without replacement. What is the probability that none of the cards are from the suit ♣.
Consider a test for a particular condition that 20% of the population is known to have. Both positive and negative results from the test are known to be 95% accurate. Compute the probability ofa. A
An insurance company sells a particular motor policy only to the inhabitants of the two English cities, Birmingham and Leicester. Sales of the policy have been successful with 10% and 25% of the
Use Wolfram Alpha to determine the probabilities of the following events.a. A dice is rolled five times, giving a different face each time.b. A random draw of four playing cards will be two jacks, a
Interpret the following expressions and indicate whether they are true or false. a. 10Px 5Px-2Px+5.3Px+7 b. 109x59x-59x+5
Calculate the following using the data in Table 9.3.a. The total number of deaths between ages 60 and 65.b. The probability that a life aged 56 exact will survive to age 80 exact.c. The probability
Use standard notation to express the following probabilities.a. The probability that an individual alive on his 50th birthday will survive a further 20 years.b. The probability that an individual
Express the following probabilities in terms of standard notation.a. The probability that a life currently aged 27 exact will die while aged 90.b. The probability that a new-born will die while aged
Using appropriate 1-year survival probabilities, construct an expression for the probability that a life aged 30 exact dies within 5 years.
Use Table 9.3 to confirm the expression for 5q30 determined in Example 9.28.Example 9.28Using appropriate 1-year survival probabilities, construct an expression for the probability that a life aged
Calculate the following probabilities using Table 9.3.a. The probability that an individual currently aged 44 exact dies while aged either 50 or 51.b. The probability that an individual currently
If A, B, and C are matrices of the same dimension, simplify 4(A + 4B+C) + 3(2B - A) − 4(5(2B+ A - C) — 2(A + 2B − C))
State, with reasons, whether the following pairs of matrices are equal: 4 4 -11 3 1-1 a. A = b. C = c. E = B = 121 254 4 3 1 -1 4 1 12 121 [134]). D=[#] 23 23 14 121 F= * = [2²4]
A matrix is said to be skew symmetric if ST = −S. Let A be any square matrix.a. Show that A − AT is skew symmetric.b. Determine matrices C and D such that A = C + D and C is symmetric and D is
You are given the following matrices:Wherever possible, obtain the following:a. D + Eb. D + Fc. E + Fd. D − F 1 -2 2 2 - 0 1 1 1 D F= 啊! 0 2 1 -3 1 2 3 -1 E= 144 422 1 2 3 (10.6)
Determine A such that A2 = 0n in each of the following situations:a. A is a general 2 ×2 matrix.b. A is a symmetric 2 × 2 matrix.
Where possible, determine the unknown matrix that solves the following matrix equations. a. b. 2 2(BT-21₂)-¹ = [i-1] 2 1 (BT-212)¹ = [81] 0
Use “aij” notation to prove the results given in Eqs. (10.3), (10.4), and (10.7). A + B = B+ A (10.3)
Ifdetermine A−1. Confirm that your result is indeed the inverse. A = 1212 1012 2121 0012
If k = 2, l = 3,compute the following expressions:a. (k + l)Ab. kA + lBc. 1/kAd. 2lA − 4kB A 12 [121 2] 21 -[ and B = 01 32
Solve the following system of linear simultaneous equations using matrix methods. x+2y+z+a = 4 x-2y+3z = 2 2x+y a = -3 a+z=2 -
Use Cramer’s rule to confirm your answer to Question 10.6.Question 10.6Solve the following system of linear simultaneous equations using matrix methods. x+2y+z+ x-2y+3z = 2 2x+y a = -3 a+z=2 a = 4 -
Use “aij” notation to prove the following properties of transposition. In each case, verify the property with a simple example using 2 × 3 matrices.a. (AT)T = Ab. (kA)T = kATc. (A + B)T = AT +
A no-claims discount model has four states and associated 1-step transition probabilities as shown in Figure 10.2. Calculate the probability that a policyholder initially in State 1 will be in State
Prove that if C and D are symmetric matrices of equal dimensions, then C + D is also symmetric. Demonstrate this property with a simple example using 4 × 4 matrices.
Calculate all possible products of two matrices that exist from C= [0 153], D= 1 4 -1 2 -3 R= [2³]. E F=[01]
You are given that the states in Question 10.8 correspond to• State 1: 0% discount• State 2: 10% discount• State 3: 40% discount• State 4: 70% discountIf the undiscounted premium is $600 per
Ifcompute the products CD and DC. C: 1234 5678 and D= 12 34 56 78
A frog moves home to a narrow strip of pond along which there are 5 evenly spaced lily pads. From each lily pad, the frog has an equal chance of jumping to the pad on either side until it reaches the
Use “aij” notation to prove that for appropriately sized matricesDemonstrate this with a simple example of rectangular matrices. (AB)T = BT AT (10.15)
Use matrixto confirm Eq. (10.20). 11 12 두 || A 1
Demonstrate that (ABC)−1 = C−1B−1A−1.
Use Eq. (10.18) to demonstrate the uniqueness of A−1 for an invertible m × m matrix A. A¹A = AA¹ = Im (10.18)
If P, Q, R, S, T, U, and V are invertible matrices of identical dimension, write down the following in terms of products of individual inverse matrices.a. (PQ)−1b. (PQRST)−1c. (VSPQRTVQ)−1d.
Demonstrate that Eq. (10.22) is true for an invertible 2 × 2 matrix. A 4 = [1 a11 a12. a21 922 A = 1 911 922 a12921 - -912 922 -421 911 (10.22)
You are given that Calculate the (1, 1)- and (3, 2)-elements of the minor and cofactor matrices, M(B) and C(B). B = 123 456 789
Express the Laplace expansion for the determinant of a general m × m matrix A alonga. The 2nd columnb. The (m − 1)th columnc. The 4th row
Confirm that the following matrix is invertible. B = 302 200 1 1 1
Compute the inverse of matrix B given in Example 10.19.Example 10.19Confirm that the following matrix is invertible. B = 302 200 1 1 1
Use the cofactor matrix stated in Eq. (10.29) to obtain D−1. C(D) = [cij (D)] = 2 3 0 2 -4 3 -3 -3 -3 3 3 -1 -3 -1 3 5 (10.29)
Determine c23(E) where E is the following 5 × 5 matrix: E = 30211 210 12 1 1 1 1 2 30121 10 12 1
Determine E−1 for E defined in Example 10.22.Example 10.22Determine c23(E) where E is the following 5 × 5 matrix: E = 30211 210 12 1 1 1 1 2 30121 10 12 1
Determine matrix C such that 2C- - ·[₁ {]-[₁ 12 -1 2 10 4 5 -5 - 2C
Find the matrix A that solves the equation 02 T (4¹-[2-1]) = 24-[69] AT 10 2A 01
Solve the following expression for A, where A, B, and C are matrices of equal dimension. 2A + B + 4C = −5A + 3B + 0.5C
Solve the following matrix equations for B. 01 23 a. (B-¹)T [ ] b. (28¹ - 1₂)¹ = [12] T 10 10 21 (HD) - [R] B 1 1 12 C. 10 21 d. ([1])" ' - [²2] B 12
Find the matrices that solve the following equations. 00 12 +4[83]- [1²] 03 a. 2A + A 00 + [83]B = [] 03 1 b. 2B+
Solve the following systems of simultaneous equations: a. b. 3a + 2c + d = 10 2a + b + d = 1 a+b+c+d=12 3a +c + 2s = −1 3a + 2c +2+e=2 2a +b+d+2e= 4 a+b+c+d+ 2e = 10 3a +c+2d+e=14 a+c+2d+e=4
Use Cramer’s rule to determine the values of b and c that solve the system in Example 10.29(a).Example 10.29(a)Solve the following systems of simultaneous equations: a. 3a + 2c +d 2a + b+d=
You are given that Use Wolfram Alpha to perform the following.a. Determine AB and BA.b. Compute det(A) and det(B).c. Compute A−1 directly.d. Compute A−1 from the cofactor of A. A = 12
You are given that a particular policyholder’s 14-year discount history is 1, 2, 1, 2, 3, 2, 3, 3, 2, 1, 1, 1, 2, 3 where the number refers to a particular discount state, as defined above.
Confirm that p(2)11 = 0.1 by considering all possible transitions over two time steps.
Obtain an expression for p(2)32 in terms of elements of the 1-step transition matrix P for a general Markov process. Interpret your expression.
Use the transition matrix in Eq. (10.34) to determine the probabilities of all possible transitions within the no-claims discount system over 4 years. P = 0.1 0.9 0 0.1 0 0.9 0 0.1 0.9 (10.34)
Obtain an expression for p(3)11 in terms of elements of the 1-step transition matrix P for a general Markov process. Interpret this expression.
An insurance company has 1000 policyholders distributed over the three states of the no-claims discount process with transition matrix given by Eq. (10.34). The initial distribution is,• 100 in
Determine whether there is a long-term, stable distribution of the 1000 policyholders inExample 10.37.Example 10.37An insurance company has 1000 policyholders distributed over the three states of the
Use implicit derivatives to determine an explicit expression for y′(x) in the following cases a. Y y² + x² = 4 b. y = cos(3x + 7y) C. 2/2 +=3y² = 2x²y³ d. exy-
Determine all tangent lines of the following implicit function at x = 1 2 x 4² + y² = 2 - xy 2 =
Classify the following ODEs. d+f=0 b. g' + xg = 2x c. x²h² +h(4) = sin x d. + y cos z = el d² dz a.

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