Poisson and Binomial Distributions

Date: 02/22/99 at 01:29:53
From: jill
Subject: Poisson and Binomial Distributions

I am just starting statistics and I am having problems understanding 
Poisson and Binomial distributions. What are the assumptions made in 
these distributions?

Date: 02/22/99 at 04:47:05
From: Doctor Mitteldorf
Subject: Re: Poisson and Binomial distributions

Think of a binomial distribution in terms of coin flips: If you flip a 
coin 3 times, you can have 0, 1, 2 or 3 heads. 0 and 3 occur 
comparatively less often than 1 and 2, because there are different 
ways to mix up the heads and tails for 1 and 2, but 0 and 3 require 
straight, consistent luck. In general, a binomial distribution 
represents the probability of getting k heads if you flip a coin n 
times.

Think of a Poisson distribution in terms of time sequence of random 
events. Imagine that you run the customer service department of a mail 
order company with millions of customers. You know that the calls come 
in at an average rate of 1 per minute. But in any given minute, you 
might have 0 calls or 1 call or 2 calls (these are pretty likely) but 
your worst nightmare is that 500 people will all decide to call in that 
same minute (unlikely, but possible). In general, you can think of a 
Poisson distribution on n and k in this way: if the AVERAGE number of 
people calling in one minute is n, what is the probability that in any 
given minute you'll get k callers?

Notice that the binomial distribution is 1) bounded at both ends and 
2) symmetric. It is bounded because you can never have more than N 
heads or less than 0. It is symmetric because the probability of 
getting k heads is always the same as the probability of getting k 
tails - which is N-k heads.

The Poisson distribution is bounded at the bottom but not the top. You 
can never have fewer than 0 calls in a given minute, but you can have 
thousands if you're really unlucky in that minute. The Poisson 
distribution is asymmetric because it's one-sided in this way.

Of course, this is only a start, but I hope it helps you to have a 
mental picture as you read more about the subject and work lots and 
lots of examples.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/