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

  • Thread starter dmatador
  • Start date
  • #1
120
1

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
124
0
Maximize/minimize = take the derivative and set it equal to zero
 
  • #3
120
1
Maybe you could elaborate on that? More specifically.
 
  • #4
124
0
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).
 
  • #5
120
1
I tried it for Y = 11, r = 5. I ended up getting p = -5, which is clearly wrong.
 
  • #6
124
0
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.
 
  • #7
120
1
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.
 

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

Replies
14
Views
5K
Replies
3
Views
19K
Replies
11
Views
3K
Replies
2
Views
933
Replies
1
Views
511
Replies
2
Views
1K
Replies
8
Views
4K
Top