# Poisson and binomial distributions, corrupted characters in a file

1. Jan 18, 2010

### Kate2010

A text file contains 1000 characters. When the file is sent by email from one machine to another, each character (independent of other characters) has probability 0.001 of being corrupted. Use a poisson random variable to estimate the probability that the file is transferred with no errors. Compare this to the answer you get when modelling the number of errors as a binomial random variable.

Poisson:
Let x be the number of corrupted characters.
E(X) = 0.001 = a
P(X=n) = (0.001^n)(e^-0.001)/(n!)
P(X=0) = e^-0.001

Binomial:
np = 1000 x 0.001 = 1
When I approximate using poisson but with np instead I get e^-1 which is about 0.37.

This doesn't seem right?

2. Jan 18, 2010

### Staff: Mentor

Show us your calculation of P(X = 0) for the binomial case.

3. Jan 18, 2010

### Kate2010

I used poisson but approximated lambda as np, so ((np)^k)(e^-np)/k! where np = 1000x0.001 = 1 and k = 0, so we get (1^0)(e^-1)/0! = e^-1.

I don't know how I could use the proper binomial random variable.

4. Jan 18, 2010

### Staff: Mentor

I believe you are supposed to assume that X is a binomial r.v., where the probability for a given character being in error is .001. You want P(X=0). See this Wikipedia page for more information.

5. Jan 18, 2010

### Kate2010

Thanks, when I do it that way I get (1000 choose 0)(0.001^0)(1-0.001)^1000 = 0.3676...
So, is it when I have calculated my poisson RV that I've gone wrong and I should have done this as I was previously trying to approximate binomial?

6. Jan 18, 2010

### Kate2010

Actually, have just read up some more on Poisson distributions and I know where I've gone wrong. Thanks for all your help!