I'm more of a programmer than a statistician, so I hope this question isn't too naive.

It happens in sampling program executions at random times. If I take N=10 random-time samples of the program's state, I could see function Foo being executed on, for example, I=3 of those samples. I'm interested in what that tells me about the actual fraction of time F that Foo is in execution.

I understand that I is binomially distributed with mean F*N. I also know that, given I and N, F follows a beta distribution. In fact I've verified by program the relationship between those two distributions, which is

cdfBeta(I, N-I+1, F) + cdfBinomial(N, F, I-1) = 1

The problem is I don't have an intuitive feel for the relationship. I can't "picture" why it works.

EDIT: All the answers were challenging, especially @whuber's, which I still need to grok, but bringing in order statistics was very helpful. Nevertheless I've realized I should have asked a more basic question: Given I and N, what is the distribution for F? Everyone has pointed out that it's Beta, which I knew. I finally figured out from Wikipedia (Conjugate prior) that it appears to be Beta(I+1, N-I+1). After exploring it with a program, it appears to be the right answer. So, I would like to know if I'm wrong. And, I'm still confused about the relationship between the two cdfs shown above, why they sum to 1, and if they even have anything to do with what I really wanted to know.

  • $\begingroup$ If "what you actually wanted to know" is "the actual fraction of time that Foo is in execution," then you are asking about a Binomial confidence interval or a (Bayesian) Binomial credible interval. $\endgroup$
    – whuber
    Nov 21, 2010 at 21:12
  • $\begingroup$ @whuber: Well I've used the random-pause method of performance tuning for over 3 decades, and some other people have discovered it too. I've told people that if some condition is true on 2 or more random-time samples, then removing it would save a good fraction of time. HOW good a fraction is what I've tried to be explicit about, assuming we don't know a Bayesian prior. Here's the general flame: stackoverflow.com/questions/375913/… and stackoverflow.com/questions/1777556/alternatives-to-gprof/… $\endgroup$ Nov 21, 2010 at 21:54
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    $\begingroup$ Nice idea. The statistical assumption is that the interruption is independent of the execution state, which is a reasonable hypothesis. A binomial confidence interval is a good tool to use to represent the uncertainty. (It can be an eye-opener, too: in your 3/10 situation, a symmetric two-sided 95% CI for the true probability is [6.7%, 65.2%]. In a 2/10 situation the interval is [2.5%, 55.6%]. These are wide ranges! Even with 2/3, the lower limit is still less than 10%. The lesson here is that something fairly rare can happen twice.) $\endgroup$
    – whuber
    Nov 21, 2010 at 22:16
  • $\begingroup$ @whuber: Thanks. You're right. Something more useful is the expected value. As far as priors go, I point out that if you only see something once, it doesn't tell you much unless you happen to know the program is in an infinite (or exceedingly long) loop. $\endgroup$ Nov 21, 2010 at 22:44
  • $\begingroup$ I think all of the answers and comments have certainly been enlightening and correct, but no one really touched on the interesting equality that @MikeDunlavey put in his original post. This equality can be found on the Beta wikipedia en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function but there is no description given of why that is the case, its just stated as a property. $\endgroup$
    – bdeonovic
    Apr 3, 2019 at 20:46

7 Answers 7


Consider the order statistics $x_{[0]} \le x_{[1]} \le \cdots \le x_{[n]}$ of $n+1$ independent draws from a uniform distribution. Because order statistics have Beta distributions, the chance that $x_{[k]}$ does not exceed $p$ is given by the Beta integral

$$\Pr[x_{[k]} \le p] = \frac{1}{B(k+1, n-k+1)} \int_0^p{x^k(1-x)^{n-k}dx}.$$

(Why is this? Here is a non-rigorous but memorable demonstration. The chance that $x_{[k]}$ lies between $p$ and $p + dp$ is the chance that out of $n+1$ uniform values, $k$ of them lie between $0$ and $p$, at least one of them lies between $p$ and $p + dp$, and the remainder lie between $p + dp$ and $1$. To first order in the infinitesimal $dp$ we only need to consider the case where exactly one value (namely, $x_{[k]}$ itself) lies between $p$ and $p + dp$ and therefore $n - k$ values exceed $p + dp$. Because all values are independent and uniform, this probability is proportional to $p^k (dp) (1 - p - dp)^{n-k}$. To first order in $dp$ this equals $p^k(1-p)^{n-k}dp$, precisely the integrand of the Beta distribution. The term $\frac{1}{B(k+1, n-k+1)}$ can be computed directly from this argument as the multinomial coefficient ${n+1}\choose{k,1, n-k}$ or derived indirectly as the normalizing constant of the integral.)

By definition, the event $x_{[k]} \le p$ is that the $k+1^\text{st}$ value does not exceed $p$. Equivalently, at least $k+1$ of the values do not exceed $p$: this simple (and I hope obvious) assertion provides the intuition you seek. The probability of the equivalent statement is given by the Binomial distribution,

$$\Pr[\text{at least }k+1\text{ of the }x_i \le p] = \sum_{j=k+1}^{n+1}{{n+1}\choose{j}} p^j (1-p)^{n+1-j}.$$

In summary, the Beta integral breaks the calculation of an event into a series of calculations: finding at least $k+1$ values in the range $[0, p]$, whose probability we normally would compute with a Binomial cdf, is broken down into mutually exclusive cases where exactly $k$ values are in the range $[0, x]$ and 1 value is in the range $[x, x+dx]$ for all possible $x$, $0 \le x \lt p$, and $dx$ is an infinitesimal length. Summing over all such "windows" $[x, x+dx]$--that is, integrating--must give the same probability as the Binomial cdf.

alt text

  • $\begingroup$ I appreciate the effort. I'm going to have to really study this because it's not my "native tongue". Also, I'm seeing a lot of dollar signs and formatting stuff. Is there something I don't know about that makes it look like real math? $\endgroup$ Nov 18, 2010 at 17:17
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    $\begingroup$ @Mike See meta.stats.stackexchange.com/q/218/919 . $\endgroup$
    – whuber
    Nov 18, 2010 at 18:12
  • $\begingroup$ I'm a little slow, but you're getting through to me. The denominator of the Beta integral is exactly what I came up with. $\endgroup$ Nov 20, 2010 at 17:48
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    $\begingroup$ It's a little late, but I finally got time to sit down and re-create your argument. The key was "multinomial coefficient". I had tried figuring it out using plain old binomial coefficients and I was getting all balled up. Thanks again for a nice answer. $\endgroup$ Jan 25, 2011 at 17:29

Look at the pdf of Binomial as a function of $x$: $$f(x) = {n\choose{x}}p^{x}(1-p)^{n-x}$$ and the pdf of Beta as a function of $p$: $$g(p)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}p^{a-1}(1-p)^{b-1}$$ You probably can see that with an appropriate (integer) choice for $a$ and $b$ these are the same. As far as I can tell, that's all there is to this relationship: the way $p$ enters into the binomial pdf just happens to be called a Beta distribution.

  • 1
    $\begingroup$ I know those look almost the same, but if I substitute y for n-x, and if I take the Beta pdf and substitute x for a-1 and y for b-1 I get an extra factor of (x+y+1), or n+1. i.e. (x+y+1)!/x!/y!*p^x*q^y. That seems to be enough to throw me off. $\endgroup$ Nov 18, 2010 at 21:10
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    $\begingroup$ Maybe somebody will chime in with a full response, but in an "intuitive" explanation we can always hand-wave away constants (like $n+1$) that do not depend on the variables of interest ($x$ and $p$), but are required to make the pdf add/integrate to 1. Feel free to replace the "equality" signs with "proportional to" signs. $\endgroup$
    – Aniko
    Nov 18, 2010 at 21:37
  • $\begingroup$ Good point. I think I'm getting closer to an understanding. I'm still trying to be able to say what x tells you about the p distribution, and why those two cdfs sum to 1. $\endgroup$ Nov 18, 2010 at 21:43
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    $\begingroup$ I take a different view of "intuitive" explanations. In some cases we don't care too much about constants, but in this case the crux of the matter is to see why an n+1 appears and not an n. If you don't understand that then your "intuition" is incorrect. $\endgroup$
    – whuber
    Nov 18, 2010 at 22:49

As you noted, the Beta distribution describes the distribution of the trial probability parameter $F$, while the binomial distribution describes the distribution of the outcome parameter $I$. Rewriting your question, what you asked about was why $$P(F \le \frac {i+1} n)+P(I \le fn-1)=1$$ $$P(Fn \le i+1)+P(I+1 \le fn)=1$$ $$P(Fn \le i+1)=P(fn<I+1)$$ That is, the likelihood that the observation plus one is greater than the expectation of the observation is the same as the likelihood that the observation plus one is greater than the expectation of the observation.

I admit that this may not help intuit the original formulation of the problem, but maybe it helps to at least see how the two distributions use the same underlying model of repeated Bernoulli trials to describe the behavior of different parameters.

  • $\begingroup$ I appreciate your take on it. All the answers are helping me to think about the question and possibly understand better what I'm asking. $\endgroup$ Nov 19, 2010 at 14:56
  • $\begingroup$ I revised the question, if you care to take a look. Thanks. $\endgroup$ Nov 20, 2010 at 15:05
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    $\begingroup$ Regarding your revision: Yes, $F\sim Beta(I+1,N-I+1)$, as long as your sampling intervals are long enough that each observation is independent and identically distributed. Note that if you want to be Bayesian about it and specify a nonuniform prior distribution for what you expect the actual proportion to be, you can add something else to both parameters. $\endgroup$
    – sesqu
    Nov 21, 2010 at 11:51
  • $\begingroup$ @sesqu, could your answer be somehow related to my question here: stats.stackexchange.com/questions/147978/…? I would appreciate your thoughts on it. $\endgroup$
    – Vicent
    Apr 23, 2015 at 21:52

Summary: It is often said that Beta distribution is a distribution on distributions! But what is means?

It essentially means that you may fix $n,k$ and think of $\mathbb P[Bin(n,p)\geqslant k]$ as a function of $p$. What the calculation below says is that the value of $\mathbb P[Bin(n,p)\geqslant k]$ increases from $0$ to $1$ when you tune $p$ from $0$ to $1$. The increasing rate at each $p$ is exactly $\beta(k,n-k+1)$ at that $p$.

enter image description here

Let $Bin(n,p)$ denote a Binomial random variable with $n$ samples and the probability of success $p$. Using basic algebra we have

$$\frac d{dp}\mathbb P[Bin(n,p)=i]=n\Big(\mathbb P[Bin(n-1,p)=i-1]-\mathbb P[Bin(n-1,p)=i]\Big).$$

It has also some nice combinatorial proof, think of it as an exercise!

So, we have:

$$\frac d{dp}\mathbb P[Bin(n,p)\geqslant k]=\frac d{dp}\sum_{i=k}^{n}\mathbb P[Bin(n,p)=i]=n\Big(\sum_{i=k}^{n}\mathbb P[Bin(n-1,p)=i-1]-\mathbb P[Bin(n-1,p)=i]\Big)$$ which is a telescoping series and can be simplified as

$$\frac d{dp}\mathbb P[Bin(n,p)\geqslant k]=n\mathbb P[Bin(n-1,p)=k-1]=\frac{n!}{(k-1)!(n-k)!}p^{k-1}(1-p)^{n-k}=\beta(k,n-k+1).$$

Remark To see an interactive version of the plot look at this. You may download the notebook or just use the Binder link.


In Bayesian land, the Beta distribution is the conjugate prior for the p parameter of the Binomial distribution.

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    $\begingroup$ Yes, but why is this the case? $\endgroup$
    – vonjd
    Aug 11, 2018 at 15:53
  • $\begingroup$ It's quite mathy but here's an exhaustive answer towardsdatascience.com/… $\endgroup$
    – jbuddy_13
    Oct 8, 2021 at 21:03

Can't comment on other answers, so i have to create my own answer.

Posterior = C * Likelihood * Prior (C is a constant that makes Posterior integrated to 1)

Given a model that uses Binomial distribution for likelihood, and Beta distribution for Prior. The product of the two which generates the Posterior is also a Beta distribution. Since the Prior and Posterior are both Beta, and thus they are conjugate distributions. the Prior (a Beta) is called conjugate prior for the likelihood (a Binomial). For example, if you multiply a Beta with a Normal, the Posterior is no longer a Beta. In summary, Beta and Binomial are two distributions that are frequently used in Bayesian inference. Beta is Conjugate Prior of Binomial, but the two distributions are not a subset or superset of the other.

The key idea of Bayesian inference is we are treating the parameter p as a random variable that ranges from [0,1] which is contrary to frequentist inference approach where we are treating parameter p as fixed. If you look closely to the properties of Beta distribution, you will see its Mean and Mode are solely determined by $\alpha$ and $\beta$ irrelevant to the parameter p . This, coupled with its flexibility, is why Beta is usually used as a Prior.


Here is an intuitive explanation that works for me:

$Binomial(n, p)$:
When repeating a Bernoulli trial with $p$ probability $n$ times. The chance of exactly $k$ successes is: $$Binomial_\mathit{pmf}(\pmb{k}, n, p) = {n\choose \pmb{k}} p^{\pmb{k}} (1-p)^{n-\pmb{k}}$$

$Beta(n, k)^*$:
For a fixed $n$ and $k$, given probability $p$, calculate the probability, $p'$, of getting $k$ in the former experiment. Then multiply this $p'$ by $n+1$ to get $k'$, the most probable (interpolated) outcome if we have done the experiment with $p'$ (conceptually this is like the mode of $Binomial(n, p')$, only it allowes for non-integer values):

$$Beta_\mathit{pdf}(\pmb{p}, n, k) = \underbrace{(n+1) \overbrace{{n \choose k} \pmb{p}^k (1-\pmb{p})^{n-k}}^{p'=Binomial_\mathit{pmf}(k, n, \pmb{p})}}_{k' \approx mode(Binomial(n, p'))}$$

$\small{*}$ I'm using ${n \choose k}$ to emphasize the similarity with the $Binomial$. To get the actual $Beta$ function we need to replace $\cdot!$ with $\Gamma(\cdot+1)$, which interpolates the factorial for non-integer values.

Note 1: If $p$ is close to $k/n$, k′ is larger.

Note 2: If the parameter $n$ is larger we are more certain of the result (the concentration is higher).

Note 3: The get the common formulation for the $Beta(\alpha,\beta)$ function:
$$k \to \alpha - 1$$ $$n \to \alpha + \beta- 2$$

Note 4: When replacing $\cdot!$ by $\Gamma(\cdot+1)$, $Beta(n, k)$ is actually defined for real-valued $n$ and $k$, with ranges $n > k - 1$ and $k > -1$. We can think of things like -0.3 successes out of -1.1 Bernoulli trials as interpolations from the integer $0 \le k \le n$ cases.

Note 5: $Beta(n=0, k=0) \equiv Uniform(0,1)$

Note 6: $\int_0^1(n+1){n\choose k} p^k (1-p)^{n-k} \,dp = 1$

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    $\begingroup$ $$(n+1){n\choose k} \pmb{p}^k (1-\pmb{p})^{n-k}$$how did you derive this? $\endgroup$ May 12, 2020 at 16:17
  • $\begingroup$ @SextusEmpiricus I think Thomas Bayes derived it :). See youtu.be/UZjlBQbV1KU?t=1074 (though his story is different) $\endgroup$ May 12, 2020 at 16:32
  • $\begingroup$ This is supposed to be an intuitive explanation. Nothing more. (I did replace 1 / BetaFunc with the (N+1) * binomial coefficient, is this the issue?) $\endgroup$ May 12, 2020 at 18:11
  • $\begingroup$ "The chance of at least one of the experiments having exactly k successes is" I have difficulties to imagine how that equals $n+1$ time the probability of having exactly k success in a single run of the experiment. For instance when p=0.5, n=2 and k=1 then the probability equals 1.5 which makes no sense. $\endgroup$ May 12, 2020 at 19:21
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    $\begingroup$ So it is a likelihood and can be above 1. But still where does the multiplication with a factor n+1 come from. Why is it not something like $$1-\left(1- {n\choose k} \pmb{p}^k (1-\pmb{p})^{n-k}\right)^{n+1}$$ according to $$p(\text{at least one event in k trials}) = 1 -(1 - p(\text{one event in one trial}))^k$$ $\endgroup$ May 12, 2020 at 20:27

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