I have to chuckle at this. I have three different sources which all disagree on what the mean of the negative binomial distribution is:
ACTEX Flash Card: E[X] = rp / q
Wikipedia article: E[X] = rq / p
Digital Actuarial Resources Study Manual: E[X] = r / p
I also own a copy of Actuarial Mathematics, published by the SOA, which ought to be more authoritative than any of the three sources above, and they say agree with Wikipedia: E[X] = rq / p.
So that ought to settle it, but I am left wondering if the differences might be because there is more than one version of the negative binomial distribution (much as there is more than one version of the geometric distribution: the regular and the "shifted").
Are these discrepancies due to typos only?
Or are there various negative binomials out there?
Or do some people use a mean that is shifted by 1?
John Strong (firstname.lastname@example.org)
Last edited by pluviosilla; August 2nd 2008 at 10:08 AM.
Reason: algebra error
Check out this EXAMPLE from the DAR Study Manual. It is interesting, because the DAR formula for the mean seems to work and the SOA / Wiki formula does not!! This convinces me that I have misunderstood the notation of these formulas.
A football kicker is making practice field goal attempts. The probability of making a field goal is 85%. The kicker will continue making field goal attempts until he makes 3 goals. What is the expected number of kicks required for the player to make 3 field goals?
Using DAR's formula, we get: E[X] = r / p = 3 / 0.85 = 3.5294
This makes perfect sense!!
On the other hand, if we use the formula in Wiki or SOA's Actuarial Mathematics, we get: E[X] = rq / p = 3*(0.15) / (0.85) = 0.529411765
This makes NO sense.
Although there are differences in notation, EVERYONE agrees on the definitions of p (probability of success) and r (number of successes), so all of these formulas for E[X] ought to be the same!
Apparently, most people define the mean of the number of *failures* before r successes, so in the example above, SOA predicts 0.5294 failures and DAR predicts 3.5294 trials to obtain 3 successes. Same difference!
Last edited by Puoya; August 2nd 2008 at 02:35 PM.
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