Answers to Section 28 of Marcel B. Finan's "A Probability Course for the Actuaries"

These answer keys are meant to assist students using Marcel B. Finan's A Probability Course for the Actuaries. With Dr. Finan's permission, Mr. Stolyarov wrote solutions for the problems in his study guide and is endeavoring to make the answer keys to each section publicly available. You can see his full List of Answer Keys here. Do the problems at the end of each section and then check your answers with these keys.

Dr. Finan's study guide is an excellent resource for those preparing to take Actuarial Exam 1/P on probability.

Problems that require numerical answers are answered here, but it is still the responsibility of the student to provide his or her own work for these problems. These answers are meant to enable students to independently verify the correctness of their reasoning by checking to see if the end result they obtained is correct. Questions from the study guide that require proofs or diagrams are not addressed here, as the end result of those questions is known in advance, and it is the responsibility of the student to provide the procedure for getting there.

Section 28

Answer 28.1. f_Y(y)= [1/{√(2π)a}]exp{−([y-b]/a−μ)²/2}

Answer 28.2. f_Y(y)= 2(y+1)/9, -1 ≤ y ≤ 2, 0 otherwise.

Answer 28.3. f_Y(y)=(1/6)y^-1/3, 0 ≤ y ≤ 8, 0 otherwise.

Answer 28.4. f_Y(y) = λy^−λ-1, y ≥ 1, 0 otherwise.

Answer 28.5. f_Y(y)= √(2)cy^1/2 m^-3/2e^−2βy/m, y ≥ 0

Answer 28.6. f_Y(y)= e^-y for 0 < y < ∞, 0 otherwise

Answer 28.7. f_Y(y)= 1/[π√(1-y²)] for − 1 ≤ y ≤ 1, 0 otherwise.

Answer 28.8a. f_Y(y)=(1/α)y^(-α+1)/α, for 0 < y < 1, 0 otherwise; E(Y)= 1/[α+1]

Answer 28.8b. f_C(c)= e^c for -∞< c < 0, 0 otherwise; E(C)=-1

Answer 28.8c. f_R(r)= 1/r for 1< r < e, 0 otherwise; E(R)= e-1

Answer 28.8d. f_N(n)= 2/[π√(1-n²)] for 0 < n < 1, 0 otherwise; E(N)=2/π

Answer 28.9. about 998.7196821

Answer 28.10. f_Y(y)=4y^-2 for y > 4, 0 otherwise.

Answer 28.11. F_V(v) =25ln(v/10000)-1 for v є (10000e^0.04, 10000e^0.08),

0 for v ≤ 10000e^0.04,

1 for v ≥ 10000e^0.08

Answer 28.12. f_Y(y)= 0.125(0.1y)^0.25exp{-(0.1y)^1.25}, for y > 0, 0 otherwise.

Answer 28.13. f_R(r)= 5/(2r²) for 5/4< r < 5/6, 0 otherwise.

Answer 28.14. f_B(b)= f_A(b/2)/2

Answer 28.15a. about 0.383

Answer 28.15b. f_Y(y)= [1/{2y√(2π)}]exp{−( ln(y)−1)²/8}

Answer 28.16. f_Y(y) =[1/B(1/2, 1)] (y)^1/2-1(1-y)^1-1, which is the pdf of a beta random variable with parameters (1/2, 1).

Answer 28.17a. f_Y(y) = (1/18)y² − 1/3, -3 ≤ y ≤ 3, 0 otherwise

Answer 28.17b. (3/2)z²- 9z + 25/2, 2 ≤ z ≤ 4, 0 otherwise

Answer 28.18. f_Y(y) = y^-1/2-1, 0 < y < 1, 0 otherwise

Answer 28.19. e^2yexp{−(e^2y)/2}, y є {Real Numbers}

See a list of all the answer keys to Dr. Finan's study guide for Exam 1/P here.