Answer:
Expectation = $409.365
Standard Deviation= $491.72.
Explanation:
Solution Let M be the amount of the medical expense and let X be the insurance company’s payout. Then,
X = {M − 100, if M > 100,
0, if M ≤ 100,
where M is exponentially distributed with parameter 1/500. To find the expected payment, apply the law of total expectation, giving
E(X) = E(E(X|M)) = ∫∞ 0 E(X/M = m)e^−m dm
= ∫
∞ 100 E(M − 100/M = m) 1
/500e^−m/500 dm
= ∫
∞
100 (m − 100) 1
/500e^−m/500 dm
= 500e^−100∕500
= $409.365.
For the standard deviation, first find
E (
X²)
= E (
E (
X²/M)) = ∫
∞ 0 E (
X²/M = m
)
e^−m dm
= ∫
∞ 100 E ( (M − 100)
²/M = m
) 1
/500e^−m/500 dm
= ∫
∞ 100 (m − 100)
² 1
/500e^−m∕500 dm
= 500000e^−1/5 = 409365.
This gives
SD(X) = √
Var(X) = √
E(X²) − E(X)²
=
√
409365 − (409.365)²
= $491.72.
NB: ∫
∞ 100 and ∫
∞ 0 is ∫ superscript ∞ and subscript 0 or 100 as the case may be.
Also, 1
/500e^−m/500 is 1/500e raised to -m/500
