) 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

i would say their habit of breaking down.

if it is for a job, then you should learn everything you can about the job, like what does it involve, what qualification is required and how you are the right person for that job. dress appropriately for the interview. get plenty of rest the night before. good luck!