If you have any shortcuts to save time during the test, please share. Here are a few of mine:

1)

This one saves from messy integration by parts, which I discovered on accident...

If a policy limit L is applied to an EXPONENTIAL distribution, the new mean is the old mean (theta) multiplied by the probability that X<L.

If a deductible D is applied to an exponential distribution, the new mean is the old mean multiplied by the probability that X>D.

These formulas, shown below, can also be used to find the deductibles or limits needed to alter the means.

Let u=original mean (theta); v=new mean after L or D is enforced...

v=u(1-e^(-L/x))

v=u(e^(-D/x))

Thus if we want the expected claim payment to be 20% less than the $500 mean damage distributed exponentially by adding either a limit or deductible...

New expected claim payment = 400

400 = 500(1-e^(-L/500))

Thus L = $804.72

OR

400 = 500(e^(-D/500)

Thus D = $111.57

+++++++++++++++++

2)

If X and Y follow independent exponential distributions with means 2 and 3, what is the probability Y<X? (And for other multivariate distributions...)

If the answers are separated by more than 3% or so, instead of double integrating the joint density function, a quick glance at the normal distribution table seems appropriate, no?

(I'm assuming that this table is the one and only information that will be provided during the test???? Anyone know?)

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