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I recently downloaded a statistical distribution application on my android, and it lists the CDF for the Binomial Distribution as $I_{1-p}(n-k,1+k)$. I am not familiar with this notation, and I was hoping someone can explain it to me.

Russell
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  • There is no simple expression for the CDF of the binomial distribution, and so the notation means whatever the originator meant it to be. For $0 \leq x \leq n$, the CDF of the binomial random variable with parameters $(n,p)$ is given by $$F_X(x) = P{X \leq x} = \sum_{i=0}^{\lfloor x \rfloor} \binom{n}{i}p^i (1-p)^{n-i}$$ and I suppose one could call this $I_{1-p}(n-k,1+k)$ if one chose to since the thing is a function of $1-p, n$, and $k$. – Dilip Sarwate Jan 07 '14 at 17:04

1 Answers1

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The notation refers to the regularized incomplete beta function.

$$I_x(a,b) = \frac{B(x; a,b)}{B(a,b)},$$

where $B(x;a,b) = \int_0^x t^{a-1}(1-t)^{b-1}\ dt$ and $B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$.

It is not particularly hard to derive this directly from the definition.

Source, for more reference: http://en.wikipedia.org/wiki/Binomial_distribution#Cumulative_distribution_function

Emily
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  • all I was looking for was $I_{1-p}(n-k,1+k)$= $(n-k)\binom{n}{k}\int_0^{1-p} t^{n-k-1}(1-t)^{k}\ dt$ – Russell Jan 07 '14 at 18:12
  • Which is not hard to get, simply by substitution. Note however, that the equality you type is not an identity, simply a consequence of that choice of values for $a,b$, and as such the right hand side is not an adequate definition of the notation $I_{1-p}\ldots$, as it is not generally true for non-integer $a,b$. – Emily Jan 07 '14 at 18:16
  • The questionable identity I typed came directly from the link you were kind enough to provide with your answer. Wiki seems to equate the two expressions, but I think I will stick with the traditional definition for the CDF of the Binomial. $$F_X(x) = P{X \leq x} = \sum_{i=0}^{\lfloor x \rfloor} \binom{n}{i}p^i (1-p)^{n-i}$$ – Russell Jan 07 '14 at 18:25
  • Yes, it is true that the things are equal (in the case where the parameters are integers), but you asked what the notation $I_{1-p}(\cdot)$ meant. – Emily Jan 07 '14 at 18:39
  • And I sincerely appreciate your response. – Russell Jan 07 '14 at 18:42
  • No problem. In case you're wondering why the CDF is presented in terms of Beta functions, versus the more straightforward summation, it's (partly) because the terms appear explicitly in the derivation of the binomial distribution's conjugate prior distribution. For instance, see the example here: http://en.wikipedia.org/wiki/Conjugate_prior_distribution. Also, beta functions are used for all sorts of things, and it's often easier/more stable numerically to compute the beta functions than the binomial coefficients in the summation. – Emily Jan 07 '14 at 18:49