ahixyz 发表于 2010-7-21 10:44
147. The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of d, the insurance company reduces the expected claim payment by 10%. Calculate the percentage reduction on the variance of the claim payment.
(A) 1%
(B) 5%
(C) 10%
(D) 20%
(E) 25%

147. Key: A
Let X denote the amount of a claim before application of the deductible. Let Y denote the amount of a claim payment after application of the deductible. Let
k be the mean of X, which because X is exponential, implies that k^2 is the variance of X and E(X^2)=2k^2. By the memoryless property of the exponential distribution, the conditional distribution of the portion of a claim above the deductible given that the claim exceeds the deductible is an exponential distribution with mean k. Given that E(Y)=0.9k, this implies that the probability of a claim exceeding the deductible is 0.9 and thus E(Y^2)=0.9*2k^2=1.8k^2.

我不太理解为什么 E(Y^2)=0.9*2k^2=1.8k^2。麻烦大家赐教~