# Questions tagged [binomial-distribution]

The binomial-distribution tag has no usage guidance.

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### Simple anticoncentration bound for binomially distributed variable

The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \...

0
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1
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37
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### Expectation of beta function with binomial distribution

Is there any way to express $E[\log(B(a+S_n,b+n-S_n))]$ where $B$ stands for beta function and $S_n \sim B(n,p)$ has a binomial distribution, in a nice way (without using multiple sums by direct ...

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91
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### Approximation of a random sum of random variables (infinitely divisible distribution) by a triangular array

We know that a Poisson distribution can be approximated by a binomial distribution. More exactly, let $(X_{jn})_{1\leq j \leq n}$ be a i.i.d. triangular array such that
$$P[X_{jn}= 1 ] = p_n = 1- P[X_{...

5
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3
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### The relative error of approximating a binomial

Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...

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### Log of a truncated binomial

Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...

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472
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### Does the (normalized) product of two independent binomial variables converge in distribution to a normal variable?

(I asked this question on MSE 10 days ago, but I got no answer.)
Let $X$ and $Y$ be two independent identically distributed binomial random variables with parameters $n \in \mathbb{N}$ and $p \in (0,1)...

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841
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### Series involving power of the index

How to prove the following identity
$$ \sum_{n=1}^{\infty} \frac{n^{n-1} e^{-n}}{n!} = 1$$
analytically (which can be confirmed with $Mathematica$)? The standard trick for geometrical series does not ...

2
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### Bound on the MGF of the product of two independent binomial, one being centered

Consider the following: $r_1,...,r_t$ are iid symmetric signs taking value $\pm1$, independent of $B\sim Binomial(p, q)$ with integer $p$ and $q=p^{-1.01}$.
Question: Consider $t$ as a non-decreasing ...

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1
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### Distribution of weight of special type of random-matrix vector product?

Let $G$ be a matrix of dimension $k \times n$ sampled uniformly randomly from $F_2^{k \times n}$. It is a well known fact that $y = xG$ is uniformly distributed in $F_2^n - \{0\}$ for all $x \in F_2^k$...

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### The largest $\ell_p$-norm of a sum of rows of a Sylvester-Hadamard-Walsh matrix

Given any $n\in\mathbb N$, consider the the Sylvester-Hadamard-Walsh matrix $M=(a_{i,j})_{i,j\in 2^n}$ of size $2^n\times 2^n$ and for a number $p\in[1,\infty)$, let
$$\nu_{n,p}=\max_{F\subseteq 2^n}\...

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### Probability of winning a $k$-rounds coin toss game

Let $p,q \in [0,1]$ with $p>q$. I denote by $B_k(p), B_k(q)$ two independent random variables following the binomial distribution, with parameters $(k,p)$ and $(k,q)$ respectively.
Informal ...

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### References on precise Large Deviations Principle/Laplace method for binomial sum

I am looking for an estimate of the following sum/expectation:
\begin{align*}%$
J_n & = \mathbb{E}\left( e^{n f(X_n) + \log(n) g(X_n) + h(X_n)} \right) \\
& = \frac{1}{2^n} \sum_{k = 0}^n {...

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### Distribution of a random variable obtained by chaining distribution

Let us define $M_0=2^n$ for $n\in\mathbb{N}$. Let $\ell\in\mathbb{N}$ be the number of random variables we are working with. For $1\leqslant i\leqslant\ell$, we define $M_i$ to be a random variable ...

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### Obtaining the error term of binomial distribution's entropy from the differential entropy of a Gaussian distribution

It is known that the first order error term in the Shannon entropy formula for a binomial distribution is $1/n$ (for example, see the Wikipedia page Binomial distribution), where in the limit $n \to \...

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### Distribution of a two-part sampling process

I have a known distribution $f(x)$ (in fact, I can safely assume that $f(x)$ is the Maxwell-Boltzmann distribution, i.e. $f(x)\propto x^2 \exp(-x^2)$). I take $N$ samples from the distribution, but am ...

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### Expected value of Taylor series with central moments of binomial variate

I want to understand this entry, but do not understand how the $\mathcal{O}\left(\frac{1}{n^2}\right)$ in the accepted answer comes into play.
I reproduce the question here: We have $x \sim \mathrm{...

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### Simmons' inequality on binomial random variables

Fix two positive integers $n,m$ such that $n>2m$ and $n\ge 3$, and let $X\sim \text{Bin}(n,\frac{m}{n})$ be a binomial random variable. For each $i\in \{1,\ldots,m\}$, set
$\alpha_i = \mathbb{P}(X\...

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### Prove zero slope point is global maximum for constrained function with binomials. Without restriction, objective function is non-concave

How to prove the zero slope point is a global maximum in this non-concave program for a function with binomials?
I need to find the (global) maximum of the following constrained problem:
$$\max_{CAP} \...

4
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1
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158
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### How to prove that these partial binomial sums are zero?

I am trying to prove that the following equation is equal to zero.
$$
0=
\sum_{j=J+1}^N \Big(j (1-q)+ (j-J) (q N-j) \Big) \cdot q^{j} (1-q)^{N -j} \binom{N}{j} \label{zero1}$$
Where
$J,N \in \...

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### CDF's of Two Binomial Distributions with the Same Mean and Different Probabilities

Suppose we have two binomial random variables $X_i \sim B(\frac{a}{p_i},p_i)$ and $X_j \sim B(\frac{a}{p_j},p_j)$, where $a$ is a positive integer, and both $\frac{a}{p_i}$ and $\frac{a}{p_j}$ are ...

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### Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$

Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...

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### better lower (and upper) bound for $i$'s moment of function of binomial random variable with $i = \frac{1}{j}, j \in \mathbb{N}$

I want to derive a lower bound for $E\left[\left(\frac{X}{k-X}\right)^{i}\right] $ with $X \sim Bin_{(k-1),p}$ and $ k \in \mathbb{N} $. So far I could prove that
\begin{equation}
E\left[\frac{X}{k-X}\...

2
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1
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185
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### sign of odd central moments of binomial distribution

I am interested in the sign of odd, central moments of a binomial distribution. From DOI. 10.1137/070700024 I have the formulae:
$ E\left[\left(X-\mu\right)^d\right]= \sum_{i=0}^n\binom{n}{i}\left(-p\...

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### Estimating the alternating sum $\sum_{j \ge 1} (-1)^j e^{-j^2} j^k$

I have been trying to get a lower bound on the following alternating sum but without much success:
$$
\sum_{j=1}^T (-1)^j e^{-j^2} j^k .
$$
For small values of $k$, this is easy because the first term ...

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### Upper bounding the sum with hypergeometric and binomial probabilities

Could you please help me upper bound this tricky expression:
$$P(A)=\sum_{i=0}^n{\left( 1 - \dfrac{\binom kq \binom {n-k}{i-q}}{\binom {n}{i}} \right)}^I \binom ni p^i {(1-p)}^{n-i}$$.
So far I only ...

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195
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### Lowerbounding expectation value of binomial tail

I'm trying to find a lower bound for the following expression for $q\ge p$:
$$f(q,p,n) := \sum_{v=0}^n \sum_{k=v}^n \binom{n}{v} \binom{n}{k}q^v(1-q)^{n-v}p^k(1-p)^{n-k}.$$
It can be thought of as the ...

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### Bounding the probability that two binomials are equal

Note: This question was migrated from this earlier post, where it initially appeared. Following suggestions, I moved this into its own question.
Let $B_{n,p}$ denote the usual binomial random ...

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### Inequality for difference of consecutive atom probabilities for binomial distribution

Edit: This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its ...

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### What is limiting behavior for the median of $2^X+2^Y$, where $X$ and $Y$ are iid binomial variables?

Suppose $X_n$ and $Y_n$ are iid variables, each distributed binomially as $B(n,1/2)$.
Let $M_n$ be the median of $2^{X_n}+2^{Y_n}$. Empirically
$$\sqrt{2}\, \le \left(\frac{M_n}{2}\right)^{1/n}\!\! \...

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### Ensemble averaging in a random graph (or network) in the large $N$ limit [closed]

I have a random graph/network described by the adjacency matrix $(a_{ij})_{N\times N}$ where $a_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta_i=\...

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1
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### In which cases $E(e^{t S_n S_m})$ converges to $E(e^{t X Y})$

Suppose that $S_n$ and $S_m$ are two random binomial variables, which are independent and with the same distribution parameter $p$. I am wondering, in which cases $E(e^{t S_n S_m})$ converges to $E(e^...

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### A conjecture on 'truncated joint moments' of binomial coefficients under binomial distribution

This is similar in spirit to Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution but gives some total estimates. Though the other ...

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### Sum of squares of middle binomial sums or 'Truncated mean' of binomial coefficients under binomial distribution

$\mu=1+\epsilon$ where $\epsilon>0$ holds.
1.Is there a good bound for $$T=\frac{\sum_{i=-\sqrt{\mu n\ln n}}^{\sqrt{\mu n\ln n}}\binom{n}{\frac n2 +i}^2}{2^n}?$$
This quantity can be ...

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0
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### Properties of a function based on binomial distribution

$f_{n,p}(k)$ is the probability mass function of a binomial distribution with parameters $n$ and $p$ i.e, for $k \in \{0,1,2, \cdots,n\}$, $f_{n,p}(k) = \binom{n}{k}p^k(1-p)^{n-k}$. Let $F_{n,p}$ be ...

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### Tail bound regime for Binomial distribution in concentration paper

In paper 'Concentration Inequalities and Martingale Inequalities:A Survey' gives the following inequality:
My question is whether the inequality holds in regime $\lambda$ being $o(\sqrt n)$ (say $\...

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1
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139
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### Distribution of non-overlapping words in randomly generated text

The question can be described in the following way:
Suppose I have a finite language $\mathcal{L}$ over alphabet $\Sigma$.
I have a string that is composed of a concatenated series of $n$ instances ...

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### Strange inequality relating Binomial pmf and cdf

I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...

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1
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### How far do I have to go for the tail of a binomial distribution with small $p$ to be $O(1/n)$?

Let $n$ be a large integer, $p$ be a small number (say, $p=C/n$ for some constant $C \ll n$), and consider the tail of the binomial distribution $B(n,p)$, after $T$:
$$
\delta = \sum_{s=T}^{n} p^s (1-...

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502
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### The mean E(X) of negative binomial distribution [closed]

What I know about the mean of the negative binomial distribution is E(x)=r(1-p)/p. but there are some questions use E(x)=r/p as the mean. Very confusing and I don't understand at all.
For example:
...

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### Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$

Disclaimer. Question moved from SE.
Setup
Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$.
Question
What is a good upper-bound for $\mathbb E[|X-np|^r]$ ?
Solution for small $r$
If $r=2$, then ...

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### Can the sum of identically distributed dependent Bernoulli trials be binomially distributed?

If you have $n$ identically distributed Bernoulli trials whose sum is binomially distributed random variable, does it then follow that the $n$ Bernoulli trials are independent?

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### Rate of convergence for difference between conditional and marginal probability

Suppose $X\sim \text{Bin}(2n,p)$ and $X_1,X_2\sim\text{Bin}(n,p)$ are independent, with $X_1+X_2=X$. I'm interested in the rate of convergence for the absolute difference
$$
\left\vert P(X>c|X_1\...

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### Showing $o(1)$ convergence for ratio of successive binomial tail probabilities

For a Binomial$(n,p)$ random variable $X$, I'm interested in showing that
$$
\frac{P(X>c)}{P(X>c-1)}=1-o(1)
$$
uniformly in $c\in\mathcal{R}$, where $\mathcal{R}$ is the range of interest (Note ...

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### Distributions of "sequential" binomials

I have come across the following stochastic process which seems very elementary, although I do not know any appropriate terminology for it; I greatly appreciate any suggestions!
Suppose I am given ...

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### Bivariate Poisson-Binomial distribution

Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p_1,\ldots,\,p_{100}$. These probabilities are not necessarily equal each other. Consider the following random ...

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502
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### What is the probability for a Binomial to be greater than other?

Let $X = B(n, p)$ and $Y = B(k, q)$ be two random variables with binomial distribution and, let $s$ be a positive integer. Assume that $n > k+s$ and $np \geq kq+s$.
What is the probability for $X$ ...

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### Show that interval of maximum probability grows no faster than $\sqrt{n}$ for binomial distribution

Let $X \sim \text{Binom}(n, p)$ a binomial random variable. I want to show that : $$\forall 0 < t < 0.9, \quad \exists C, \quad \forall n >1, \quad \mathbb P\bigg(|X-np| \leq C\sqrt{n}\bigg) \...

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### Total Variation distance of polynomials of Bernoulli R.V.s

Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with
$\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$.
Let
\begin{align*}
X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...

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### Lower bound for the probability that $X=\omega\left(\mathbb E[X]\right)$ for $X\sim Bin(n,p)$

Let $X\sim Bin(n,p)$ be a binomial variable and let $\delta\in (0,1)$.
I'm looking for a lower bound of the form $\Pr[X > f(\delta)] \ge \delta$.
Specifically, if $\delta,p=o(1)$ are not ...

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### Tartaglia distribution

I do not know if this question is elementary of advanced. Let me start by describing the two dimensional case in word.
Take the Tartaglia (or Pascal, or Bernoulli, or whatever name you want to give ...