Math Is Fun Forum

George23

Guest

Probability Question

If anyone could explain how the following is done, it would be greatly appreciated!

Compute the expectation of X when X has a negative binomial distribution.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Probability Question

Here's some fine reading that got me to understand it.
http://stattrek.com/Lesson2/NegBinomial.aspx
or click
http://stattrek.com/Lesson2/NegBinomial.aspx
If that didn't do it for you, then you
probably you should practice comparing binary numbers with combinations of letters.
For example,
pick two letters out of ABCD where order doesn't matter.
six ways: AB AC AD BC BD CD
Now label columns and use binary to check off things in the columns.
ABCD
1100 = AB
0011 = CD
1001 = AD
0110 = BC
1010 = AC
0101 = BD
This may seem dumb, but it is neat to
see that the permuations of 0011 where order
does matter is equivalent to the 2 letters chosen out of 4, where order does not matter.

This brings us to making a diagram of the success and failures
in a repeated trial.
Say that on the 7th trial, 2 successes were found.
The formula says you use 6C1 (times probabilities) and this is why.
Track out the possibilities with columns being trial repetitions:
S=success  F=failure
123456| 7
SFFFFF | S
FSFFFF | S
FFSFFF | S
FFFSFF | S
FFFFSF | S
FFFFFS | S
If chance of any independent trial failure F = 23% and success S = 77%,
then FFFSFFS is chance of (23%^5)(77%^2).
And so for 7 trials and 2 successes, the last trial always being a success you see,
then [(7-1)C(2-1)](23%^5)(77%^2) is your expectation mayperbehaps, if I understand what I just read. 
Hope I'm on the right track for you because I just learned this for you.
Bye.

Last edited by John E. Franklin (2007-10-04 18:20:53)

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