) A health insurance will pay for a medical expense subject to a $100 deductible. Assume that the amount of the expense is exponentially distributed with mean $500. Find Find the expectation and standard deviation of the payout by the insurance company.

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

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