Actuary Certification ACTUARY Mathematics and Statistics 2 — Questions and Answers
Question 1: A random variable X follows a Poisson distribution with mean λ = 4. What is P(X = 2)?
- 0.0916
- 0.1465 (Correct answer)
- 0.1954
- 0.2381
Correct answer: 0.1465
P(X=2) = e^(-4) * 4^2 / 2! = e^(-4) * 8 ≈ 0.1465.
Question 2: Which of the following is the correct formula for the variance of a binomial distribution with parameters n and p?
- np
- np(1-p) (Correct answer)
- np²
- n²p(1-p)
Correct answer: np(1-p)
The variance of a binomial distribution is np(1-p), where n is the number of trials and p is the success probability.
Question 3: If X and Y are independent random variables with Var(X) = 5 and Var(Y) = 3, what is Var(2X - Y + 7)?
- 11
- 17
- 23 (Correct answer)
- 27
Correct answer: 23
Var(2X - Y + 7) = 4·Var(X) + Var(Y) = 4(5) + 3 = 23; constants have zero variance.
Question 4: The moment generating function of a standard normal distribution is:
- e^t
- e^(t²/2) (Correct answer)
- e^(μt + σ²t²/2)
- 1/(1-t)
Correct answer: e^(t²/2)
For Z ~ N(0,1), the MGF is M(t) = e^(t²/2).
Question 5: An actuary fits a linear regression model Y = β₀ + β₁X + ε. The coefficient of determination R² = 0.81 and SST = 200. What is SSE (sum of squared errors)?
- 38 (Correct answer)
- 162
- 200
- 242
Correct answer: 38
SSE = SST(1 - R²) = 200(1 - 0.81) = 200(0.19) = 38.
Question 6: For a continuous uniform distribution on [a, b], what is the variance?
- (b-a)/2
- (b-a)²/4
- (b-a)²/12 (Correct answer)
- (b-a)/12
Correct answer: (b-a)²/12
The variance of U(a,b) is (b-a)²/12.
Question 7: If the joint density of (X,Y) is f(x,y) = 2 for 0 < x < y < 1, what is the marginal density of Y?
- 2y (Correct answer)
- 2(1-y)
- 1
- 2y(1-y)
Correct answer: 2y
f_Y(y) = ∫₀ʸ 2 dx = 2y for 0 < y < 1.
Actuary Certification ACTUARY Mathematics and Statistics 2
A random variable X follows a Poisson distribution with mean λ = 4.
What is P(X = 2)?