Exponential growth and exponential distribution

[jimmy1]

I'm a bit confused about exp growth and exp distribution. Suppose I have a branching process situation, where there are n individuals in generation 0, and each individual produces a random number of offspring, according to some distribution (say Poisson), at each generation. Now, then after a certain number of generations, say generation n+1, the number of copies an individual leaves behind will be either 0 or exponentially distributed.

What I don't understand is how the number of copies will be exponentially distributed. I understand that if each an individual leaves more than 1 offspring in each generation then there will be exponential growth for that particular type of individual, but isn't exponential growth different from the exponential distribution.
From my understanding an exponential distribution gives the distribution of the time until a certain event occurs, with rate "lamda".

So in the situation described above what is this "event", and what should "lamda" be??

So basically I'm trying to understand, how does the exponential distribution model the number of offspring left by an individual??

 

[EnumaElish]

I think one could start from asking what is random about the population growth you proposed. Is it a binary proposition, 0=no offspring, 1=positive number of offsprings? Is it multi-valued, number of offspring = 1, 2, ..., each outcome having a probability attached to it? Is it the number of offsprings within a time frame (e.g. lifespan of the parent)?

 

[jimmy1]

Basically the number of offspring an inidividual leaves behind is a random number, based on a Poisson distribution. It could be 0, 1, 2 ... etc. (You are given the mean offspring number)

Say you have 10 populations, and each population starts off with 1 person. Everyone then dies (at the same time), and leaves a Poisson distributed number of offspring, (which is the next generation). These offspring then die and again each person leaves a Poisson distributed number of offspring. This process continues for n generations.
So after n generations each population will be at some random number of individuals. Now the distribution of these numbers after n generations, I am told follows an exponential distribution?? And this is what I don't really understand, as in how does the exponential distribution describe the number of offspring left, as I taught the exponential distribution only describes waiting times for a certain event??

(Also the mean offspring number does not change throughout the whole process)

 

[EnumaElish]

The only relation I can see with the exp. dist. is the memoryless property. Exp. dist. is the only continuous dist. with (and is characterized by) the property that its graph over [t, infinity) is identical to its graph over [0, infinity) for any t > 0. In your example, if you remove the first m generations, the distribution of the offspring emanating from anyone individual in anyone population after g generations starting with the m+1st gen. will be identical to the distribution of the offspring emanating from the original Adam or Eve in that population after g generations starting with the 1st gen. (Wouldn't it?) I am not sure that this is the answer; but it is the only connection that I can make to the exp. dist.

I guess the growth pattern in your question may be said to be fractal, in that sense.