Answers to Section 30 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 30

Answer 30.1a. X and Y are independent.

Answer 30.1b. ½= 0.5

Answer 30.1c. 1-e^−a

Answer 30.2c. π/4= about 0.7853981634

Answer 30.3a. 0.484375= 31/64

Answer 30.3b. f_X(x)= 3x²-6x+3, for 0 ≤ x < 1, 0 otherwise

f_Y(y)=6y(1-y) for 0 < y ≤ 1, 0 otherwise

Answer 30.3c. X and Y are not independent.

Answer 30.4a. k=4

Answer 30.4b. f_X(x)= 2x, 0 ≤ x ≤ 1, 0 otherwise

f_Y(y)= 2y, 0 ≤ y ≤ 1, 0 otherwise

Answer 30.4c. X and Y are independent.

Answer 30.5a. k=6

Answer 30.5b. f_X(x)=2x, 0 ≤ x ≤ 1, 0 otherwise

f_Y(y)= 3y², 0 ≤ y ≤ 1, 0 otherwise

Answer 30.5c. 0.15= 3/20

Answer 30.5d. 0.875= 7/8

Answer 30.5e. X and Y are independent.

Answer 30.6a. k=8/7

Answer 30.6b. X and Y are independent.

Answer 30.6c. 16/21= about 0.7619047619

Answer 30.7a. f_X(x)= x²/4+1/6, 0 ≤ x ≤ 2, 0 otherwise

f_Y(y)= 1/3+y/6, 0 ≤ y ≤ 2, 0 otherwise

Answer 30.7b. X and Y are not independent.

Answer 30.7c. 31/72= about 0.43055555555555555556

Answer 30.8a. f_X(x)= 4/3-8x/9, 0 ≤ x ≤ 3/2, 0 otherwise.

f_Y(y)= 4y/9 for 0 ≤ y ≤ 1.5,

4/3-4y/9 for 1.5 ≤ y ≤ 3,

0 otherwise.

Answer 30.8b. 2/3

Answer 30.8c. X and Y are not independent.

Answer 30.9. about 0.4691535082

Answer 30.10. about 0.1914554452

Answer 30.11. 2/5= 0.4

Answer 30.12. 0.19= 19/100

Answer 30.13. 0.295

Answer 30.14. about 0.4137812139

Answer 30.15. f_Z(z)= e^−z/2-e^−z, z > 0

Answer 30.16. f_X(x)= 2/(1+2x)²

Answer 30.17. P(X² ≥ Y³)= 3/5 = 0.6

Answer 30.18. It is impossible that X and Y are independent, as .4 = P(X = 1|Y = 1) ≠ P(X = 1|Y = 0) = 0.7.

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