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?)

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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