ACTUARY Mathematics and Statistics Flashcards
7 cards from real Actuary Certification practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.
Read the first 7 ACTUARY Mathematics and Statistics flashcards as text
The Central Limit Theorem states that, for a large sample of size n drawn from a population with mean μ and variance σ², the sample mean X̄ is approximately:
Answer: N(μ, σ²/n)
By the CLT, X̄ ≈ N(μ, σ²/n) for large n regardless of the population distribution.
An actuary calculates that the limited expected value E[X ∧ d] = 800 for a loss variable X with mean E[X] = 1200. What is the expected value of the excess loss above d?
Answer: 400 × S(d)
E[X] = E[X ∧ d] + E[(X-d)₊], and E[(X-d)₊] = (E[X] - E[X ∧ d]) is split as (1200-800) = 400 times the survival function at d.
Which of the following distributions has heavier tails than the normal distribution?
Answer: Pareto
The Pareto distribution has heavy (power-law) tails, much heavier than the normal, making it common in actuarial loss modeling.
If Cov(X, Y) = 6, Var(X) = 9, and Var(Y) = 16, what is the correlation coefficient ρ(X,Y)?
Answer: 0.5
ρ = Cov(X,Y) / [SD(X)·SD(Y)] = 6 / (3 × 4) = 6/12 = 0.5.
A fair six-sided die is rolled three times. What is the probability of getting exactly two sixes?
Answer: 15/216
P = C(3,2)·(1/6)²·(5/6) = 3·(1/36)·(5/6) = 15/216.
In the context of credibility theory, which formula represents the Bühlmann credibility estimate?
Answer: Z·X̄ + (1-Z)·μ
The Bühlmann credibility estimate is Z·X̄ + (1-Z)·μ, where Z = n/(n+k) is the credibility factor.
Which of the following is a property of an unbiased estimator θ̂ for parameter θ?
Answer: E[θ̂] = θ
An estimator is unbiased if its expected value equals the true parameter value: E[θ̂] = θ.