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
