Help with negative binomial distributions

• Mar 25th 2010, 03:46 PM
mintsharpie
Help with negative binomial distributions
One of the questions in my probability homework reads:

X denotes a negative binomial random variable, with p = 0.6 Find P(X ≥ 3) for a) r = 2 and b) r = 4.

According to my teacher, the answers are 0.1792 and 0.45568, respectively, but I can't for the life of me figure out how he got them. I tried finding P(X ≥ 3) by turning it into 1 - P(X ≤ 2) and then calculating p(2), p(1), and p(0), but I kept getting 0 for my answer, which obviously isn't correct.

Thanks.
• Mar 25th 2010, 03:57 PM
Random Variable

$P(X=0) = \binom{1}{1} 0.6^{2}0.4^{0} = 0.36$

$P(X=1) = \binom{2}{1} 0.6^{2}0.4^{1} = 0.288$

$P(X=2) = \binom{3}{1} 0.6^{2}0.4^{2} = 0.1728$

$1 - 0.36-0.288-0.1728 = 0. 1792$
• Mar 25th 2010, 04:18 PM
mintsharpie
Quote:

Originally Posted by Random Variable

$P(X=0) = \binom{1}{1} 0.6^{2}0.4^{0} = 0.36$

$P(X=1) = \binom{2}{1} 0.6^{2}0.4^{1} = 0.288$

$P(X=2) = \binom{3}{1} 0.6^{2}0.4^{2} = 0.1728$

$1 - 0.36-0.288-0.1728 = 0. 1792$

Thanks so much, this helped a lot. But I'm still a little confused about where the 3 came from in the P(X=2) equation. For future reference, how would I know how to find the number that goes there (same for the numbers in the first bracket of P(X=1) and P(X=0)?

Edit: I also don't understand part b. I don't understand which numbers go where. Can you please help me?
• Mar 26th 2010, 02:14 PM
mr fantastic
Quote:

Originally Posted by mintsharpie
Thanks so much, this helped a lot. But I'm still a little confused about where the 3 came from in the P(X=2) equation. For future reference, how would I know how to find the number that goes there (same for the numbers in the first bracket of P(X=1) and P(X=0)?

Edit: I also don't understand part b. I don't understand which numbers go where. Can you please help me?

I don't see what the problem can be here. Unless you have never seen the pmf for the negative binomial distribution, in which case you are advised to consult your textbook.

The pmf is given here: Negative binomial distribution - Wikipedia, the free encyclopedia

and all you do is substitute the given numbers into it. Where do you get stuck in doing this?
• Mar 26th 2010, 03:50 PM
mintsharpie
Quote:

Originally Posted by mr fantastic
I don't see what the problem can be here. Unless you have never seen the pmf for the negative binomial distribution, in which case you are advised to consult your textbook.

The pmf is given here: Negative binomial distribution - Wikipedia, the free encyclopedia

and all you do is substitute the given numbers into it. Where do you get stuck in doing this?

Thank you so much! I didn't understand what the pmf was before. Sorry for double-posting, I just wanted an answer. But you've helped me figure it out now, so thanks. (Happy)