An insurance company sells two types of auto insurance policies: Basic and Deluxe. The time until the next Basic Policy claim is an exponential random variable with mean 6 days. The time until the next Deluxe Policy claim is an independent exponential random variable with mean 7 days. What is the probability that the next claim will be a Deluxe Policy claim?

a) 0.462

b) 0.562

c) 0.362

d) 0.529

e) 0.329

I know how to do this but I can't get the double integral to work out...

As I understand it:

x - time until Basic claim

y - time until Deluxe claim

Find P(y < x)

since the two functions are independent, we have:

f_XY(x,y) = f_X(x) * f_Y(y) = (1/6)e^(-x/6) * (1/7)e^(-y/7)

then you would just double integrate f(x,y) from

int(0 to infinity) int(0 to x) of f_XY(x,y) dydx

I don't know why I am having such a hard time integrating this... I came up with the answer .4487...

Is there a trick to solving this other than just brute force. Any chance you can use the Survival function? :embarrassed:

a) 0.462

b) 0.562

c) 0.362

d) 0.529

e) 0.329

I know how to do this but I can't get the double integral to work out...

As I understand it:

x - time until Basic claim

y - time until Deluxe claim

Find P(y < x)

since the two functions are independent, we have:

f_XY(x,y) = f_X(x) * f_Y(y) = (1/6)e^(-x/6) * (1/7)e^(-y/7)

then you would just double integrate f(x,y) from

int(0 to infinity) int(0 to x) of f_XY(x,y) dydx

I don't know why I am having such a hard time integrating this... I came up with the answer .4487...

Is there a trick to solving this other than just brute force. Any chance you can use the Survival function? :embarrassed:

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