An exponential cumulative probability distribution has the following
form: F(x) = 1-exp(-x/m)

What is the probability distribution?
What is the mean and the variance of the distribution?
What is the expected value of exp(x) for such a distribution and what is it called?
The exponential distribution is often assumed as representing a ?time
between events? distribution.  In this case, how would you count the
number of events that occur (for example, the number of claims made by
an insured) over a period of time T?  Just specify the logical
procedure.
If each event has a loss associated to it with a cumulative
probability distribution G(L), how would you simulate the loss that
the individual has over a period of time T?

Hello

   Researcher usually ignore multiple questions in one 'GA question',
particularly for low or mid priced questions like this one .

I am making an exception in the hope that
 you will be able to find all answers in few selected links below.

It should required only a limited effort.

1) p.d.f. is exponential distribution    P=(1/m) * exp (-x/m) 
   it is shown on Figs 2, 3, here:
http://cnx.org/content/m13128/latest/
where you will find basic properties (mean, etc)

2) waiting times for events distributed uniformly have indeed
exponential distribution. Probability that n such event will happen in
a given time
 is given by Poisson distribution.

http://en.wikipedia.org/wiki/Poisson_distribution

All this including

3) simulation

is covered by Queue theory. Most textbooks start with example of simples queue
which uses all these concepts you mention and gives examples.

http://people.bu.edu/pekoz/feed.pdf

more referrences:

http://www.answers.com/topic/poisson-process
http://www.ise.canberra.edu.au/un6538/Lectures/2006/Poisson.pdf
http://en.wikipedia.org/wiki/Queuing_theory

http://www2.uwindsor.ca/~hlynka/qonline.html

To focus on simulation (as a tool for solving more complex situation) use

SEARCH TERM: discrete event simulation

e.g.
http://www.mathworks.com/access/helpdesk/help/toolbox/simevents/ug/index.html?/access/helpdesk/help/toolbox/simevents/ug/bp8wu4e.html


Hedgie

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Request for Answer Clarification by mathdumbie-ga on 29 Oct 2006 08:38 PST

Thank you, this is perfect.  In regards to my last bulleted item, "If
each event has a loss associated to it with a cumulative probability
distribution G(L), how would you simulate the loss that the individual
has over a period of time T" could you please point me in the right
direction as well (or did I just miss it)?

---

Clarification of Answer by hedgie-ga on 29 Oct 2006 21:59 PST

Well, I did not know which c.d.f.  the G(L)  is. 

a)  Are we still talking about
 uniform-expo-poisson case defined by F(x) and m 

or is (entirely new) topic, namely 

b) For any given c.d.f. G(L) what is the AVERAGE cost rate C  (e.g.
$/day) if cost  of each event is  c1  ?

  I assume here  a stationary (steady state) process, so all I ned is P(n)
   probability that n events happanes in a day (=interval T). Then

 C = (SUM over n) ( P(n) * n*c1 )

 In case a) P(n) is the Poisson distribution  with mean determined from F(x)
 For general G(L) it will not be Poisson, but probability Px(n) than n 
 will happen N trials can be derived same way. See

SEARCH TERM bernoulli  probability distribution, binary 

e.g. 
http://www.statsoft.com/textbook/stdisfit.html

Poisson is a special case for case of rare events


Rating appreciated

Good Luck with your studies

Hedgie

Excellent resource!