**cab@allegra.UUCP (Carol Beck)** (09/27/84)

Probability is not my forte, and I'm stuck on what seems like a simple problem. Here goes. I've derived a probability density function for a continuous random variable, t, where t is the time duration of some event. What I need is the probability density function of the random variable which is the maximum time duration of n events. (i.e. I start n events at the same time. What is the pdf of the time at which the last of these n events completes?) Is there some insight I'm missing that makes this intuitively obvious? Any suggestions welcome. Thanks, Carol Beck allegra!cab

**eklhad@ihnet.UUCP (K. A. Dahlke)** (09/29/84)

< munch munch salavate slurp chomp > Given the probability of an event completing at time t, what is the probability of the last (out of n) events completing at time t? Let me first clarify terms, since they are often used ambiguously. Let me use the "density" function (p(t)) to indicate the probability of an event completing near time t. and the "distribution" function (P(t)) to indicate the probability of the event completing before time t. Thus P(t) = integral(x=0,t) p(x). P(t) starts at 0 and monotonically increases to 1. I have seen text books use the terms "density" and "distribution" this way, and two chapters later, describe the distribution of a random variable with a density function (e.g. uniform distribution: a nice horizontal line). But I digress. At time t1, the event has already completed with probability P(t1). I am assuming multiple events are independant. If two events are started together, their times will both be less than t1 with probability P(t1)*P(t1). Simply raise P(t) to the nth power for n events. This produces the composite distribution function. If you need the composite density function, differentiate the result. You can then use the density function to obtain mean, variance, etc. -- Karl Dahlke ihnp4!ihnet!eklhad

**leimkuhl@uiucdcsb.UUCP** (10/01/84)

Actually, we were not told that the durations are iid. Are they?