How to maximize P(Y = y*) for a negative binomial distribution

[dmatador]

How can I find probability p that maximized P(Y = y*) when Y has a negative binomial distribution with parameters r (known) and p? I've just reduced the problem with some algebra, but other than guess-and-check I have no rigorous way to solve this problem.

 

[diggy]

Maximize/minimize = take the derivative and set it equal to zero

 

[dmatador]

Maybe you could elaborate on that? More specifically.

 

[diggy]

Sorry was just trying to avoid using latex.

But write down your binomial dist. It should be
neg_binomial = A * p^k * (1-p)^r
where A won't depend on p, but will be some combinatoric.

Now take the derivative with respect to p.
A*k*p^(k-1)*(1-p)^r + A*r*p^k*(1-p)^(r-1)
now set that equal to 0 and solve for p. It will be some explicit value depending on your parameters.

Sorry for skimping on the latex, if its too unclear I can re-write it for you.

edit: But the point is that whenever you want to find a max or min, that is an equivalent statement to saying that the slope of the function has to be zero (although a zero slope isn't always a max or min).

 

[dmatador]

I tried it for Y = 11, r = 5. I ended up getting p = -5, which is clearly wrong.

 

[diggy]

I tried it for Y = 11, r = 5. I ended up getting p = -5, which is clearly wrong.

First off, one obvious mistake on what I wrote to you before.

The second term of the expression should be negative not positive (from the chain rule on the derivative of (1-p)). Maybe double check that the rest is right.

If it still doesn't give a sensible answer I'll work it out explicitly tomorrow.

Sorry about that.

 

[dmatador]

Oh no worries, it works now. But I guess you'll have to keep q in the form 1 - p for this to work, because you won't get a minus sign between the two terms.