2009 年 9月18日SOA Exam P机经

1)
There are 15 people, 10 are government officials and 5 are non-government officials. Suppose a 4 people committee would be formed. Find the probability of having AT LEAST 3 government officials in the committee.

(Hypergeometric distribution)

Answer: Simply plug in the pmf formula for hypergeometric distribution

Let n be number of government officials inside the committee.

P (n = 3) = (10C3 * 5C1) / 15C4 = 0.43956
P (n = 4) = (10C4) / 15C4 = 0.1538
So that, P (n >= 3) = 0.43956 + 0.1538 = 0.5934

2)
Suppose a machine has two parts, X and Y, the Y would be functioning after X has break down, and the machine would be out of order if Y has broken down (after X’s failure). The joint probability density function of X and Y is given by:

f(x,y) = e ^ (0.8x + 0.4y), x > 0 and y > 0, where x and y follow exponential distribution.

Find the expected life (in years) of X + Y.

Answer:
Using mean of exponential distribution function, 1 / 0.8 + 1 / 0.4 = 3.75 (years)

3)
The probability density function (f(x)) of X is given by

| x | / 10 for -2 < x < 4
0 otherwise

Find the expected value of x.

Answer:

Decompose | x | into 2 parts,
If -2 < x < 0, | x | = - x
0 < x < 4, | x | = x

Then using E(x) = integration of x * f(x) with appropriate bounds can find the answer easily.


4)

Suppose there are three independent events, namely X1 X2 X3. They all follow exponential distribution with mean 2. Find the probability that the maximum of these 3 events would not exceed a constant, e.g. K.

Solution flow: Using F(a) = 1 – P (X > a) can easily solve the problem.

Since X1 X2 X3 are independent events, as long as the every event would not exceed K, then the maximum of them would not exceed K.

So that:
P (Max (x1,x2,x3) <= K) = P(X1 <=K) * P (X2 <=K) * P (X3 <= K) (independence).

5)

Use normal approximation to approximate binomial distribution (with continuity correction)

6)
(partial)

The joint distribution of a couple’s life is given by

f(x,y) = e ^ ( 0.015x + 0.014y ), x and y follows exponential distribution.

Calculate: Variance of (x) given y > 25

(This is the problem of joint distribution and conditional variance)

我觉得那个郭玉峰( Guo Yufeng)的manual挺好的，缺陷就是比考试要真實深一些，

Gamma, beta, Pareto, Weibull and bivariate normal distribution 全沒出現, 可以完全不理!

另外 hypergeometric distribution 只要懂 pmf 即可.

重点是 exponential distribution, 一定要熟記 mean variance 及積分捷徑! 因為共計有九題 ( 30%) 涉及 exponential distriubution.

Marginal density / conditional density 也要用到
要畫熟 2D-region for solving joint density function.

希望對大家有用!!

心得: 我全做 SOA sample questions , 我做了 2 遍 ,真考有3 - 4 條解題思路完全一樣!

+ Guo Yufeng P Manual

(exclude chapters of Gamma, beta, Pareto, Weibull, Chi-Square AND bivariate normal distribution, joint moment generating function, Univariate & joint order statistics, Markov's inequality , "Risk and Insurance")

P.S 我以前是市塲学 (marketing) 的 , 數学底子一般也 unofficially 過了 , good luck to all!! Thanks god too!