A company offers earthquake insurance. Annual premiums are modeled by an exponential random variable and mean 2. Annual claims are modeled by an exponential random variable with mean 1. Premiums and claims are independent. Let X denote the ratio of claims to premiums.

Premium f(t) = (1/2)e^(-t/2)

Claim g(t) = e^(-t)

X = g(t)/f(t) 'the ratio?'

^anyways, that's what I was thinking... then I saw the solutions...

u = annual claims

v = annual premiums

"next you need to make a joint density function of U and V... "

g(u,v) = joint density function of U and V = [e^(-u)][(1/2)e^(-v/2)] with ranges 0<u<inf and 0<v<inf

"next you must create the distribution function of X; in order to make the pdf of X"

F(x) = Pr[X<=x] = P[u/v <= x] "ok, there's the ratio"

^this I've seen a lot, and I keep forgetting about it.

= P[U<=Vx] "I don't get this part"

What does this mean? The probability that U is less than or equal to V times X? and that equals the integral of g(u,v)dudv...... I'm stumped where this comes from.... Is there a similar problem to this that is easier to understand?? Oh, the range of U is from 0 < u < vx; that's smurfy because it's the probability of U<=Vx and also u>0.

Ok, some recap.. when solving for the distribution function and you have some function f(a,b). You may start with something like P[X<=x] or P[Y<=y]. Then, you want to solve the Pr[??<=??] to be either Pr[a<=??] -OR- Pr[b<=??] and respectively the range of a would be 0<a<?? and the range of b would be 0<b<??.

finding f(x) is somewhat easy after getting F(x)

F'(x) = f(x) = [(-1)(2x+1)^(-1) + 1]' = 2(-1)(-1)(2x+1)^(-2) = 2*(2x+1)^(-2)

1) What would you call this type of problem? Or what key words would I use to research info on this type of problem.

2) Is there an easier way to explain the steps that I'm confused on? All my explanations were derived using the solution and some logic.

Premium f(t) = (1/2)e^(-t/2)

Claim g(t) = e^(-t)

X = g(t)/f(t) 'the ratio?'

^anyways, that's what I was thinking... then I saw the solutions...

u = annual claims

v = annual premiums

"next you need to make a joint density function of U and V... "

g(u,v) = joint density function of U and V = [e^(-u)][(1/2)e^(-v/2)] with ranges 0<u<inf and 0<v<inf

"next you must create the distribution function of X; in order to make the pdf of X"

F(x) = Pr[X<=x] = P[u/v <= x] "ok, there's the ratio"

^this I've seen a lot, and I keep forgetting about it.

= P[U<=Vx] "I don't get this part"

What does this mean? The probability that U is less than or equal to V times X? and that equals the integral of g(u,v)dudv...... I'm stumped where this comes from.... Is there a similar problem to this that is easier to understand?? Oh, the range of U is from 0 < u < vx; that's smurfy because it's the probability of U<=Vx and also u>0.

Ok, some recap.. when solving for the distribution function and you have some function f(a,b). You may start with something like P[X<=x] or P[Y<=y]. Then, you want to solve the Pr[??<=??] to be either Pr[a<=??] -OR- Pr[b<=??] and respectively the range of a would be 0<a<?? and the range of b would be 0<b<??.

finding f(x) is somewhat easy after getting F(x)

F'(x) = f(x) = [(-1)(2x+1)^(-1) + 1]' = 2(-1)(-1)(2x+1)^(-2) = 2*(2x+1)^(-2)

1) What would you call this type of problem? Or what key words would I use to research info on this type of problem.

2) Is there an easier way to explain the steps that I'm confused on? All my explanations were derived using the solution and some logic.

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