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#1
02-10-2007, 02:32 AM
 achiu Join Date: Feb 2007 Posts: 11
Can anyone help to solve those questions?

I was told by my friend the following questions are from recent past exam (don't know which yr) but don't know whether this is true??? Anyway, I found they are interesting. Can anyone solve them? Thanks you!!

1. The number of severe storms that strike city J in a year follows a binomial distribution with n=5 and p=0.6. Given that m severe storms strike city J in a year, the number of severe storms that strike city K in the same year is m with probability 1/2, m+1 with probability 1/3, m+2 with probability 1/6.
Calculate the expected number of severe storms that strike city J in a year during which 5 severe storms strike city K.

2. The random variable Y1=e^X1 characterizes an insurer's annual property losses, where X1 is normally distributed with mean 16 and standard deviation 1.5. Similarly, the random variable Y2=eX2 characterizes the insurer's annual liability losses, where X2 is normally distributed with mean 15 and standard deviation 2.
The insurer's annual property losses are independent of its annual liability losses. Calculate the probability that, in a given year, the minimum of the insurer's property losses and liability losses exceeds e^16.

3. Let X be a continuous random variable with probability density function f(x)=te^(-tx) for x>0 and t>0. Let Y be the smallest interger greater than or equal to X. Determine the probability function of Y.
#2
02-10-2007, 10:13 AM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 26,459

Quote:
 Originally Posted by achiu 1. The number of severe storms that strike city J in a year follows a binomial distribution with n=5 and p=0.6. Given that m severe storms strike city J in a year, the number of severe storms that strike city K in the same year is m with probability 1/2, m+1 with probability 1/3, m+2 with probability 1/6. Calculate the expected number of severe storms that strike city J in a year during which 5 severe storms strike city K.
I think you need to derive the conditional distribution of J (severe storms hitting city J) given K (severe storms hitting city K) = 5.
Start by computing the unconditional joint densities for the case where k = 5:

f(3,5) = 1/6 p(j=3)
f(4,5) = 1/3 p(j=4)
f(5,5) = 1/2 p(j=5)

g(5) = marginal density for k=5 = sum of those

f(j|5) = f(j,5)/g(5), for j=3,4,5

E(J|5) = sum j=3 to 5 of j f(j|5)

Quote:
 2. The random variable Y1=e^X1 characterizes an insurer's annual property losses, where X1 is normally distributed with mean 16 and standard deviation 1.5. Similarly, the random variable Y2=eX2 characterizes the insurer's annual liability losses, where X2 is normally distributed with mean 15 and standard deviation 2. The insurer's annual property losses are independent of its annual liability losses. Calculate the probability that, in a given year, the minimum of the insurer's property losses and liability losses exceeds e^16.
When is the minimum less than e^16? It happens unless Y1 and Y2 are each > e^16.
Pr(Y1>e^16) = Pr(X1>16)
Pr(Y2>e^16) = Pr(X2>16)
With a standard normal distribution table, you should be able to determine each of those.

Independent, so multiply to get the probabilty both are greater. Subtract result from 1 to get probability at least one is less (so that the minimum is less)

Quote:
 3. Let X be a continuous random variable with probability density function f(x)=te^(-tx) for x>0 and t>0. Let Y be the smallest interger greater than or equal to X. Determine the probability function of Y.
One way is to start by getting the cumulative distribution function of X, F(x). Then for any integer k, the cumulative probability function for k is
P(k) = F(k) (P(k) exists only for integers).
Then the probability function is P(k) - P(k-1).

Alternatively, what values of X correspond to K=k? Ans: k-1 < x <= k
What is the probability X is in that interval: integral from k-1 to k of te^(-tx) dx.
#3
02-13-2007, 10:13 AM
 achiu Join Date: Feb 2007 Posts: 11

Thank you so much!
#4
02-13-2007, 07:10 PM
 Generalshamu Member SOA Join Date: Oct 2006 Location: LA Favorite beer: Heineken Posts: 811

Quote:
 Originally Posted by Gandalf When is the minimum less than e^16? It happens unless Y1 and Y2 are each > e^16. Pr(Y1>e^16) = Pr(X1>16) Pr(Y2>e^16) = Pr(X2>16) With a standard normal distribution table, you should be able to determine each of those. Independent, so multiply to get the probabilty both are greater. Subtract result from 1 to get probability at least one is less (so that the minimum is less)
Wait don't you have to do X = ln Y here, since it follows a lognormal distribution???
From that you get Phi((ln x - mean )/standard deviation)?
For my answer I got .15425???

-----

Just for clarification, is the answer to the first question, 3.897637795?

Last edited by Generalshamu; 02-13-2007 at 07:16 PM..
#5
02-13-2007, 07:20 PM
 Colymbosathon ecplecticos Member Join Date: Dec 2003 Posts: 4,988

Quote:
 Originally Posted by Generalshamu Wait don't you have to do X = ln Y here, since it follows a lognormal distribution??? From that you get Phi((ln x - mean )/standard deviation)? For my answer I got .15425??? ----- Just for clarification, is the answer to the first question, 3.897637795?
Recall: Y1 = exp(X1). When is Y1>exp(16)?

Right, when X1 > 16, because exp is monotone increasing.
#6
02-13-2007, 07:34 PM
 Generalshamu Member SOA Join Date: Oct 2006 Location: LA Favorite beer: Heineken Posts: 811

Oh but you get the same Z-scores of you do it this way...is it wrong to take it as a lognormal distribution?
#7
02-13-2007, 07:42 PM
 atomic Member CAS Join Date: Jul 2006 Posts: 4,088

Quote:
 Originally Posted by Generalshamu Oh but you get the same Z-scores of you do it this way...is it wrong to take it as a lognormal distribution?
Because the logarithmic and exponential functions are order-preserving, one-to-one mappings, it's not necessary to deal with the lognormal distribution of Y. It suffices to simply look at the normal distribution of X. That is to say, the minimum of Y1 and Y2 is the same as the exponential of the minimum of X1 and X2, or

$\min\{Y_1, Y_2\} = \min\{e^{X_1}, e^{X_2}\} = \exp(\min\{X_1, X_2\})$.
__________________
Spoiler:
"No, Moslems don't believe Jesus was the messiah.

Think of it like a movie. The Torah is the first one, and the New Testament is the sequel. Then the Qu'ran comes out, and it retcons the last one like it never happened. There's still Jesus, but he's not the main character anymore, and the messiah hasn't shown up yet.

Jews like the first movie but ignored the sequels, Christians think you need to watch the first two, but the third movie doesn't count, Moslems think the third one was the best, and Mormons liked the second one so much they started writing fanfiction that doesn't fit with ANY of the series canon."

-RandomFerret
#8
02-13-2007, 08:11 PM
 Generalshamu Member SOA Join Date: Oct 2006 Location: LA Favorite beer: Heineken Posts: 811

Can you do that with ANY density function?

Where if they both have the same factors, you can just reduce the densities accordingly?
#9
02-13-2007, 08:35 PM
 atomic Member CAS Join Date: Jul 2006 Posts: 4,088

Quote:
 Originally Posted by Generalshamu Can you do that with ANY density function? Where if they both have the same factors, you can just reduce the densities accordingly?
I'm not sure what you mean by "reduce the densities accordingly."

If your question is whether the content of my previous post depends on the density of X1 and X2, the answer is no; all that is required is that the same order-preserving, one-to-one mapping be applied to each distribution. The key relationship is that g(a) < g(b) if and only if a < b, for a, b in the support of X1, X2, ..., Xn. Then min{g(X1), g(X2), ..., g(Xn)} = g(min{X1, X2, ..., Xn}).

If I may ask, is your background in mathematics? What I find curious is that you seem to have a propensity for asking questions that imply you are wanting to find very general yet powerful rules for working with probability distributions--things that you can apply in every case, while avoiding dealing with first principles. In my opinion, this is an unwise approach to understanding mathematics. Rather, a broad, solid foundation in the basics of mathematical reasoning is what allows one to develop "shortcuts" while still being aware of when they can be applied, or how they arise. I realize that this sort of learning is not the most time-efficient approach for the exams, but it is again in my view the best approach over the long run.
__________________
Spoiler:
"No, Moslems don't believe Jesus was the messiah.

Think of it like a movie. The Torah is the first one, and the New Testament is the sequel. Then the Qu'ran comes out, and it retcons the last one like it never happened. There's still Jesus, but he's not the main character anymore, and the messiah hasn't shown up yet.

Jews like the first movie but ignored the sequels, Christians think you need to watch the first two, but the third movie doesn't count, Moslems think the third one was the best, and Mormons liked the second one so much they started writing fanfiction that doesn't fit with ANY of the series canon."

-RandomFerret
#10
02-13-2007, 08:47 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 26,459

This is sort of saying what atomic did, and relating to my earlier post about looking at x instead of y. There are lots of situations where you have a condition like g(x) > k, or k1 < g(x) < k2, etc.

In those situations, it's often a good idea to try to figure out which values of x would make the inequality involving g(x) true. It tends to be easiest to do that if g(x) is a 1-to-1 increasing function, but the idea of figuring out what values of x work can be helpful in other situations, too.

E.g., if you want to know where g(x) = x^2 > 4, it is true when x < -2 or x > 2. Often that is a productive way to attack the problem.

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