# Binomial distribution

(Redirected from Binomial probability)
"Binomial model" redirects here. For the binomial model in options pricing, see Binomial options pricing model.
Notation Probability mass function Cumulative distribution function B(n, p) n ∈ N0 — number of trials p ∈ [0,1] — success probability in each trial k ∈ { 0, …, n } — number of successes $\textstyle {n \choose k}\, p^k (1-p)^{n-k}$ $\textstyle I_{1-p}(n - k, 1 + k)$ $np$ $\lfloor np \rfloor$ or $\lceil np \rceil$ $\lfloor (n + 1)p \rfloor$ or $\lfloor (n + 1)p \rfloor - 1$ $np(1 - p)$ $\frac{1-2p}{\sqrt{np(1-p)}}$ $\frac{1-6p(1-p)}{np(1-p)}$ $\frac12 \log_2 \big( 2\pi e\, np(1-p) \big) + O \left( \frac{1}{n} \right)$ in shannons. For nats, use the natural log in the log. $(1-p + pe^t)^n \!$ $(1-p + pe^{it})^n \!$ $G(z) = \left[(1-p) + pz\right]^n.$ $g_n(p) = \frac{n}{p(1-p)}$ (for fixed $n$)
Binomial distribution for $p=0.5$
with n and k as in Pascal's triangle

The probability that a ball in a Galton box with 8 layers (n = 8) ends up in the central bin (k = 4) is $70/256$.

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. A success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution is a good approximation, and widely used.

## Specification

### Probability mass function

In general, if the random variable X follows the binomial distribution with parameters n ∈ ℕ and p ∈ [0,1], we write X ~ B(np). The probability of getting exactly k successes in n trials is given by the probability mass function:

$f(k;n,p) = \Pr(X = k) = \binom n k p^k(1-p)^{n-k}$

for k = 0, 1, 2, ..., n, where

$\binom n k =\frac{n!}{k!(n-k)!}$

is the binomial coefficient, hence the name of the distribution. The formula can be understood as follows: we want exactly k successes (pk) and n − k failures (1 − p)n − k. However, the k successes can occur anywhere among the n trials, and there are ${n\choose k}$ different ways of distributing k successes in a sequence of n trials.

In creating reference tables for binomial distribution probability, usually the table is filled in up to n/2 values. This is because for k > n/2, the probability can be calculated by its complement as

$f(k,n,p)=f(n-k,n,1-p).$

The probability mass function satisfies the following recurrence relation, for every $n,p$:

$\left\{\begin{array}{l} p (n-k) f(k,n,p) = (k+1) (1-p) f(k+1,n,p), \\[10pt] f(0,n,p)=(1-p)^n \end{array}\right\}$

Looking at the expression ƒ(knp) as a function of k, there is a k value that maximizes it. This k value can be found by calculating

$\frac{f(k+1,n,p)}{f(k,n,p)}=\frac{(n-k)p}{(k+1)(1-p)}$

and comparing it to 1. There is always an integer M that satisfies

$(n+1)p-1 \leq M < (n+1)p.$

ƒ(knp) is monotone increasing for k < M and monotone decreasing for k > M, with the exception of the case where (n + 1)p is an integer. In this case, there are two values for which ƒ is maximal: (n + 1)p and (n + 1)p − 1. M is the most probable (most likely) outcome of the Bernoulli trials and is called the mode. Note that the probability of it occurring can be fairly small.

### Cumulative distribution function

The cumulative distribution function can be expressed as:

$F(k;n,p) = \Pr(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n\choose i}p^i(1-p)^{n-i}$

where $\scriptstyle \lfloor k\rfloor\,$ is the "floor" under k, i.e. the greatest integer less than or equal to k.

It can also be represented in terms of the regularized incomplete beta function, as follows:[1]

\begin{align} F(k;n,p) & = \Pr(X \le k) \\ &= I_{1-p}(n-k, k+1) \\ & = (n-k) {n \choose k} \int_0^{1-p} t^{n-k-1} (1-t)^k \, dt. \end{align}

Some closed-form bounds for the cumulative distribution function are given below.

## Example

Suppose a biased coin comes up heads with probability 0.3 when tossed. What is the probability of achieving 0, 1,..., 6 heads after six tosses?

$\Pr(0\text{ heads}) = f(0) = \Pr(X = 0) = {6\choose 0}0.3^0 (1-0.3)^{6-0} \approx 0.1176$
$\Pr(1\text{ heads}) = f(1) = \Pr(X = 1) = {6\choose 1}0.3^1 (1-0.3)^{6-1} \approx 0.3025$
$\Pr(2\text{ heads}) = f(2) = \Pr(X = 2) = {6\choose 2}0.3^2 (1-0.3)^{6-2} \approx 0.3241$
$\Pr(3\text{ heads}) = f(3) = \Pr(X = 3) = {6\choose 3}0.3^3 (1-0.3)^{6-3} \approx 0.1852$
$\Pr(4\text{ heads}) = f(4) = \Pr(X = 4) = {6\choose 4}0.3^4 (1-0.3)^{6-4} \approx 0.0595$
$\Pr(5\text{ heads}) = f(5) = \Pr(X = 5) = {6\choose 5}0.3^5 (1-0.3)^{6-5} \approx 0.0102$
$\Pr(6\text{ heads}) = f(6) = \Pr(X = 6) = {6\choose 6}0.3^6 (1-0.3)^{6-6} \approx 0.0007$[2]

## Mean

If X ~ B(n, p), that is, X is a binomially distributed random variable, n being the total number of experiments and p the probability of each experiment yielding a successful result, then the expected value of X is:[3]

$\operatorname{E}[X] = np ,$

(For example, if n=100, and p=1/4, then the average number of successful results will be 25)

Proof: The mean μ can be directly calculated from its definition $\mu=\sum_{i=1}^n x_i p_i$ and the binomial theorem:

\begin{align} \mu &= \sum_{k=0}^n k\binom nk p^k (1-p)^{n-k}\\ &= np\sum_{k=0}^n k\frac{(n-1)!}{(n-k)!k!}p^{k-1} (1-p)^{(n-1)-(k-1)}\\ &= np\sum_{k=1}^n \frac{(n-1)!}{((n-1)-(k-1))!(k-1)!}p^{k-1} (1-p)^{(n-1)-(k-1)}\\ &= np\sum_{k=1}^n \binom{n-1}{k-1} p^{k-1} (1-p)^{(n-1)-(k-1)}\\ &= np\sum_{\ell=0}^{n-1} \binom{n-1}\ell p^\ell (1-p)^{(n-1)-\ell}\quad\text{with } \ell:=k-1\\ &= np\sum_{\ell=0}^m \binom m\ell p^\ell (1-p)^{m-\ell}\qquad\text{with } m:=n-1\\ &= np\left(p+\left(1-p\right)\right)^m=np1^m=np \end{align}

It is also possible to deduce the mean from the equation $X=X_1+X_2+\ldots+X_n$ whereby all $X_i$ are Bernoulli distributed random variables with $E[X_i]=p$. We get

$E[X] = E[X_1+X_2+\ldots+X_n] = E[X_1]+E[X_2]+\ldots+E[X_n] = \underbrace{p+p+\ldots+p}_{n\text{ times}} = np$

## Variance

The variance is:

$\operatorname{Var}[X] = np(1 - p).$

Proof: Let $X=X_1+X_2+\ldots+X_n$ with all $X_i$ be i.i.d Bernoulli distributed random variables. We get because of $\operatorname{Var}(X_i) = pq$:

$\operatorname{Var}(X)=\operatorname{Var}(X_1+\dotsb+X_n)=\operatorname{Var}(X_1)+\dotsb+\operatorname{Var}(X_n)=n \operatorname{Var}(X_1)=np\left(1-p\right)=npq$

## Mode

Usually the mode of a binomial B(n, p) distribution is equal to $\lfloor (n+1)p\rfloor$, where $\lfloor\cdot\rfloor$ is the floor function. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes: (n + 1)p and (n + 1)p − 1. When p is equal to 0 or 1, the mode will be 0 and n correspondingly. These cases can be summarized as follows:

$\text{mode} = \begin{cases} \lfloor (n+1)\,p\rfloor & \text{if }(n+1)p\text{ is 0 or a noninteger}, \\ (n+1)\,p\ \text{ and }\ (n+1)\,p - 1 &\text{if }(n+1)p\in\{1,\dots,n\}, \\ n & \text{if }(n+1)p = n + 1. \end{cases}$

Proof Let $f(k)=\binom nk p^k q^{n-k}$. For $p=0$ only $f(0)$ has a nonzero value with $f(0)=1$ and for $p=1$ we find $f(n)=1$ and $f(k)=0$ for $k\neq n$. This proves that the mode is 0 for $p=0$ and n for $p=1$.

Let $0 < p < 1$. We find $\frac{f(k+1)}{f(k)} = \frac{(n-k)p}{(k+1)(1-p)}$. From this follows

\begin{align} k > (n+1)p-1 \Rightarrow a_{k+1} < a_k \\ k = (n+1)p-1 \Rightarrow a_{k+1} = a_k \\ k < (n+1)p-1 \Rightarrow a_{k+1} > a_k \end{align}

So when $(n+1)p-1$ is an integer, then $(n+1)p-1$ and $(n+1)p$ is a mode. In the case that $(n+1)p-1\notin \Z$, then only $\lfloor (n+1)p-1\rfloor+1=\lfloor (n+1)p\rfloor$ is a mode.[4]

## Median

In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However several special results have been established:

• If np is an integer, then the mean, median, and mode coincide and equal np.[5][6]
• Any median m must lie within the interval ⌊np⌋ ≤ m ≤ ⌈np⌉.[7]
• A median m cannot lie too far away from the mean: |mnp| ≤ min{ ln 2, max{p, 1 − p} }.[8]
• The median is unique and equal to m = round(np) in cases when either p ≤ 1 − ln 2 or p ≥ ln 2 or |m − np| ≤ min{p, 1 − p} (except for the case when p = ½ and n is odd).[7][8]
• When p = 1/2 and n is odd, any number m in the interval ½(n − 1) ≤ m ≤ ½(n + 1) is a median of the binomial distribution. If p = 1/2 and n is even, then m = n/2 is the unique median.

## Covariance between two binomials

If two binomially distributed random variables X and Y are observed together, estimating their covariance can be useful. Using the definition of covariance, in the case n = 1 (thus being Bernoulli trials) we have

$\operatorname{Cov}(X, Y) = \operatorname{E}(XY) - \mu_X \mu_Y.$

The first term is non-zero only when both X and Y are one, and μX and μY are equal to the two probabilities. Defining pB as the probability of both happening at the same time, this gives

$\operatorname{Cov}(X, Y) = p_B - p_X p_Y,$

and for n independent pairwise trials

$\operatorname{Cov}(X, Y)_n = n ( p_B - p_X p_Y ).$

If X and Y are the same variable, this reduces to the variance formula given above.

## Related distributions

### Sums of binomials

If X ~ B(np) and Y ~ B(mp) are independent binomial variables with the same probability p, then X + Y is again a binomial variable; its distribution is[citation needed] Z=X+Y ~ B(n+mp):

\begin{align} \operatorname P(Z=k) &= \sum_{i=0}^k\left[\binom{n}i p^i (1-p)^{n-i}\right]\left[\binom{m}{k-i} p^{k-i} (1-p)^{m-k+i}\right]\\ &= \binom{n+m}k p^k (1-p)^{n+m-k} \end{align}

However, if X and Y do not have the same probability p, then the variance of the sum will be smaller than the variance of a binomial variable distributed as $B(n+m, \bar{p}).\,$

### Conditional binomials

If X ~ B(np) and, conditional on X, Y ~ B(Xq), then Y is a simple binomial variable with distribution[citation needed]

$Y \sim B(n, pq).$

For example imagine throwing n balls to a basket UX and taking the balls that hit and throwing them to another basket UY. If p is the probability to hit UX then X ~ B(np) is the number of balls that hit UX. If q is the probability to hit UY then the number of balls that hit UY is Y ~ B(Xq) and therefore Y ~ B(npq).

### Bernoulli distribution

The Bernoulli distribution is a special case of the binomial distribution, where n = 1. Symbolically, X ~ B(1, p) has the same meaning as X ~ Bern(p). Conversely, any binomial distribution, B(np), is the distribution of the sum of n Bernoulli trials, Bern(p), each with the same probability p.[citation needed]

### Poisson binomial distribution

The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of n independent non-identical Bernoulli trials Bern(pi).[9] If X has the Poisson binomial distribution with p1 = … = pn =p then X ~ B(np).

### Normal approximation

Binomial probability mass function and normal probability density function approximation for n = 6 and p = 0.5

If n is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to B(np) is given by the normal distribution

$\mathcal{N}(np,\,\sqrt{np(1-p})),$

and this basic approximation can be improved in a simple way by using a suitable continuity correction. The basic approximation generally improves as n increases (at least 20) and is better when p is not near to 0 or 1.[10] Various rules of thumb may be used to decide whether n is large enough, and p is far enough from the extremes of zero or one:

• One rule is that both x=np and n(1 − p) must be greater than 5. However, the specific number varies from source to source, and depends on how good an approximation one wants; some sources give 10 which gives virtually the same results as the following rule for large n until n is very large (ex: x=11, n=7752).
• A second rule[10] is that for n > 5 the normal approximation is adequate if
$\left | \left (\frac{1}{\sqrt{n}} \right ) \left (\sqrt{\frac{1-p}{p}}-\sqrt{\frac{p}{1-p}} \right ) \right |<0.3$
• Another commonly used rule holds that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values,[citation needed] that is if
$\mu \pm 3 \sigma = np \pm 3 \sqrt{np(1-p)} \in [0,n].$

The following is an example of applying a continuity correction. Suppose one wishes to calculate Pr(X ≤ 8) for a binomial random variable X. If Y has a distribution given by the normal approximation, then Pr(X ≤ 8) is approximated by Pr(Y ≤ 8.5). The addition of 0.5 is the continuity correction; the uncorrected normal approximation gives considerably less accurate results.

This approximation, known as de Moivre–Laplace theorem, is a huge time-saver when undertaking calculations by hand (exact calculations with large n are very onerous); historically, it was the first use of the normal distribution, introduced in Abraham de Moivre's book The Doctrine of Chances in 1738. Nowadays, it can be seen as a consequence of the central limit theorem since B(np) is a sum of n independent, identically distributed Bernoulli variables with parameter p. This fact is the basis of a hypothesis test, a "proportion z-test", for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic.[11]

For example, suppose one randomly samples n people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If groups of n people were sampled repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation $\sigma = \sqrt{\frac{p(1-p)}{n}}$

### Poisson approximation

The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product np remains fixed. Therefore the Poisson distribution with parameter λ = np can be used as an approximation to B(n, p) of the binomial distribution if n is sufficiently large and p is sufficiently small. According to two rules of thumb, this approximation is good if n ≥ 20 and p ≤ 0.05, or if n ≥ 100 and np ≤ 10.[12]

### Limiting distributions

$\frac{X-np}{\sqrt{np(1-p)}}$
approaches the normal distribution with expected value 0 and variance 1.[citation needed] This result is sometimes loosely stated by saying that the distribution of X is asymptotically normal with expected value np and variance np(1 − p). This result is a specific case of the central limit theorem.

### Beta distribution

Beta distributions provide a family of conjugate prior probability distributions for binomial distributions in Bayesian inference. The domain of the beta distribution can be viewed as a probability, and in fact the beta distribution is often used to describe the distribution of a probability value p:[13]

$P(p;\alpha,\beta) = \frac{p^{\alpha-1}(1-p)^{\beta-1}}{\mathrm{B}(\alpha,\beta)}$.

## Confidence intervals

Even for quite large values of n, the actual distribution of the mean is significantly nonnormal.[14] Because of this problem several methods to estimate confidence intervals have been proposed.

Let n1 be the number of successes out of n, the total number of trials, and let

$\hat{p} = \frac{n_1}{n}$

be the proportion of successes. Let zα/2 be the 100(1 − α/2)th percentile of the standard normal distribution.

• Wald method
$\hat{p} \pm z_{\frac{\alpha}{2}} \sqrt{ \frac{ \hat{p} ( 1 -\hat{p} )}{ n } } .$
A continuity correction of 0.5/n may be added.[clarification needed]
• Agresti-Coull method[15]
$\tilde{p} \pm z_{\frac{\alpha}{2}} \sqrt{ \frac{ \tilde{p} ( 1 - \tilde{p} )}{ n + z_{\frac{\alpha}{2}}^2 } } .$
Here the estimate of p is modified to
$\tilde{p}= \frac{ n_1 + \frac{1}{2} z_{\frac{\alpha}{2}}^2}{ n + z_{\frac{\alpha}{2}}^2 }$
• ArcSine method[16]
$\sin^2 \left (\arcsin \left ( \sqrt{ \hat{p} } \right ) \pm \frac{ z }{ 2 \sqrt{ n } } \right )$
• Wilson (score) method[17]
$\frac{\hat{p} + \frac{1}{2n} z_{1-\frac{\alpha}{2}}^2 \pm \frac{1}{2n} z_{1-\frac{\alpha}{2}} \sqrt{4n\hat{p}(1 - \hat{p})+ z_{1-\frac{\alpha}{2}}^2}} {1+ \frac{1}{n} z_{1-\frac{\alpha}{2}}^2}.$

The exact (Clopper-Pearson) method is the most conservative.[14] The Wald method although commonly recommended in the text books is the most biased.[clarification needed]

## Generating binomial random variates

Methods for random number generation where the marginal distribution is a binomial distribution are well-established.[18][19]

One way to generate random samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that P(X=k) for all values k from 0 through n. (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a linear congruential generator to generate samples uniform between 0 and 1, one can transform the calculated samples U[0,1] into discrete numbers by using the probabilities calculated in step one.

## Tail Bounds

For knp, upper bounds for the lower tail of the distribution function can be derived. Recall that $F(k;n,p) = \Pr(X \le k)$, the probability that there are at most k successes.

Hoeffding's inequality yields the bound

$F(k;n,p) \leq \exp\left(-2 \frac{(np-k)^2}{n}\right), \!$

and Chernoff's inequality can be used to derive the bound

$F(k;n,p) \leq \exp\left(-\frac{1}{2\,p} \frac{(np-k)^2}{n}\right). \!$

Moreover, these bounds are reasonably tight when p = 1/2, since the following expression holds for all k3n/8[20]

$F(k;n,\tfrac{1}{2}) \geq \frac{1}{15} \exp\left(- \frac{16 (\frac{n}{2} - k)^2}{n}\right). \!$

However, the bounds do not work well for extreme values of p. In particular, as p $\rightarrow$ 1, value F(k;n,p) goes to zero (for fixed k, n with k<n) while the upper bound above goes to a positive constant. In this case a better bound is given by [21]

$F(k;n,p) \leq \exp\left(-nD\left(\frac{k}{n}\left|\right|p\right)\right) \quad\quad\mbox{if }0<\frac{k}{n}

where D(a|| p) is the relative entropy between an a-coin and a p-coin (i.e. between the Bernoulli(a) and Bernoulli(p) distribution):

$D(a||p)=(a)\log\frac{a}{p}+(1-a)\log\frac{1-a}{1-p}. \!$

Asymptotically, this bound is reasonably tight; see [21] for details. An equivalent formulation of the bound is

$\Pr(X \ge k) =F(n-k;n,1-p)\leq \exp\left(-nD\left(\frac{k}{n}\left|\right|p\right)\right) \quad\quad\mbox{if }p<\frac{k}{n}<1.\!$

Both these bounds are derived directly from the Chernoff bound. It can also be shown that,

$\Pr(X \ge k) =F(n-k;n,1-p)\geq \frac{1}{(n+1)^2} \exp\left(-nD\left(\frac{k}{n}\left|\right|p\right)\right) \quad\quad\mbox{if }p<\frac{k}{n}<1.\!$

This is proved using the method of types (see for example chapter 12 of Elements of Information Theory by Cover and Thomas [22]).

We can also change the $(n+1)^2$ in the denominator to $\sqrt{2n}$, by approximating the binomial coefficient with Stirlings formula.[23]

## References

1. ^ Wadsworth, G. P. (1960). Introduction to Probability and Random Variables. New York: McGraw-Hill. p. 52.
2. ^ Hamilton Institute. "The Binomial Distribution" October 20, 2010.
3. ^ See Proof Wiki
4. ^
5. ^ Neumann, P. (1966). "Über den Median der Binomial- and Poissonverteilung". Wissenschaftliche Zeitschrift der Technischen Universität Dresden (in German) 19: 29–33.
6. ^ Lord, Nick. (July 2010). "Binomial averages when the mean is an integer", The Mathematical Gazette 94, 331-332.
7. ^ a b Kaas, R.; Buhrman, J.M. (1980). "Mean, Median and Mode in Binomial Distributions". Statistica Neerlandica 34 (1): 13–18. doi:10.1111/j.1467-9574.1980.tb00681.x.
8. ^ a b Hamza, K. (1995). "The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions". Statistics & Probability Letters 23: 21–25. doi:10.1016/0167-7152(94)00090-U.
9. ^ Wang, Y. H. (1993). "On the number of successes in independent trials" (PDF). Statistica Sinica 3 (2): 295–312.
10. ^ a b Box, Hunter and Hunter (1978). Statistics for experimenters. Wiley. p. 130.
11. ^ NIST/SEMATECH, "7.2.4. Does the proportion of defectives meet requirements?" e-Handbook of Statistical Methods.
12. ^ a b NIST/SEMATECH, "6.3.3.1. Counts Control Charts", e-Handbook of Statistical Methods.
13. ^ MacKay, David (2003). Information Theory, Inference and Learning Algorithms. Cambridge University Press; First Edition. ISBN 978-0521642989.
14. ^ a b Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban (2001), "Interval Estimation for a Binomial Proportion", Statistical Science 16 (2): 101–133, doi:10.1214/ss/1009213286, retrieved 2015-01-05
15. ^ Agresti, Alan; Coull, Brent A. (May 1998), "Approximate is better than 'exact' for interval estimation of binomial proportions" (PDF), The American Statistician 52 (2): 119–126, doi:10.2307/2685469, retrieved 2015-01-05
16. ^
17. ^ Wilson, Edwin B. (June 1927), "Probable inference, the law of succession, and statistical inference" (PDF), J. American Statistical Association 22 (158): 209–212, doi:10.2307/2276774, retrieved 2015-01-05
18. ^ Devroye, Luc (1986) Non-Uniform Random Variate Generation, New York: Springer-Verlag. (See especially Chapter X, Discrete Univariate Distributions)
19. ^ Kachitvichyanukul, V.; Schmeiser, B. W. (1988). "Binomial random variate generation". Communications of the ACM 31 (2): 216–222. doi:10.1145/42372.42381.
20. ^ Matoušek, J, Vondrak, J: The Probabilistic Method (lecture notes) [1].
21. ^ a b R. Arratia and L. Gordon: Tutorial on large deviations for the binomial distribution, Bulletin of Mathematical Biology 51(1) (1989), 125–131 [2].
22. ^ Theorem 11.1.3 in Cover, T.; Thomas, J. (2006). Elements of Information Theory (2nd ed.). Wiley. p. 350.
23. ^ http://math.stackexchange.com/questions/1548940/sharper-lower-bounds-for-binomial-chernoff-tails/1564088#1564088
24. ^ Mandelbrot, B. B., Fisher, A. J., & Calvet, L. E. (1997). A multifractal model of asset returns. 3.2 The Binomial Measure is the Simplest Example of a Multifractal