Question

A government employee’s yearly dental expense is a random
variable X with pdf

f(x) = 1 / 1000, if 200 < x < 1200; and f(x) = 0, otherwise.
The government’s primary dental plan reimburses an employee for up to 400 of
dental expense incurred in a year, while a supplemental plan pays up to 500 of
any remaining dental expense. Let Y represent the yearly benefit paid by the
supplemental plan to a government employee. Verify that E(Y) = 275 and
compute the standard deviation of Y.
Hint: Y = 0 if X does not exceed 400, and Y = min (X – 400, 500) if X exceeds 400.

Answer 1

Let X be the employee's yearly dentail expense incurred which is distributed uniformly in the interval (200,1200)

The pdf of X is

The cdf of X is

Let Y represent the benefit paid by the supplimental plan

• If X<=400 then the supplimental plan need not pay anything, or Y=0

the probability of Y=0 is the probability of X<=400 and it is

• the suppliment pays upto 500, that is if X>400, Y=X-400 and is between 0 to 500 till X=900. That is when X is between 400 and 900, Y is uniformly distributed in the interval (0,500)

the probability of X is between 400 and 900 is

The conditional pdf of Y given that X is between 400 and 900 is (Y is uniform (0,500)

The conditional cdf of Y is

The joint distribution of Y when X is between 400 and 900 is

• Finally when the claim is greater than 900, the suplimental plan Y pays only 500. The probability of Y=500 is the probability of X>900 and it is

To consolidat the probability distribution of Y is

The expected value of Y is

Next we calculate

Finally the variance of Y is

std dev =sqrt variance =sqrt(41042) =

202.5883