Questions tagged [binomial-distribution]

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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 ...
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1 vote
1 answer
48 views

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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  • 930
1 vote
1 answer
85 views

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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4 votes
1 answer
181 views

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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3 votes
0 answers
72 views

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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  • 131
3 votes
0 answers
47 views

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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  • 549
0 votes
0 answers
29 views

Bound on the minimum cell frequency of a multinomial

Is there an upper bound to the cumulative distribution of the minimum cell frequency of a multinomial distribution that is tighter than the simple union bound? Let $X \sim \mathrm{Multinomial}(n; p_1,\...
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  • 141
1 vote
1 answer
47 views

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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0 votes
1 answer
189 views

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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1 vote
1 answer
54 views

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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0 votes
2 answers
245 views

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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4 votes
0 answers
381 views

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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1 vote
1 answer
98 views

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} \...
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4 votes
1 answer
142 views

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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1 vote
1 answer
208 views

How to prove this "obvious" result from Bernoulli trials / partial Pascal series?

The main problem setup - very brief I need to prove that the following two equations have a solution with $CAP=v$. $$0= ( CAP - C ) \sum_{j=1}^J j N \cdot q^{j-1} (1-q)^{N -1-j} \binom{N}{j} ...
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1 vote
1 answer
99 views

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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1 vote
1 answer
859 views

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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  • 173
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1 answer
187 views

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}\...
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  • 91
2 votes
1 answer
102 views

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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  • 91
7 votes
0 answers
211 views

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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  • 141
0 votes
0 answers
43 views

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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  • 1
-1 votes
1 answer
159 views

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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7 votes
1 answer
495 views

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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3 votes
1 answer
175 views

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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  • 2,610
3 votes
2 answers
144 views

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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-2 votes
1 answer
50 views

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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  • 137
-2 votes
1 answer
105 views

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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1 vote
2 answers
116 views

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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  • 1,728
2 votes
1 answer
183 views

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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1 vote
0 answers
167 views

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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0 votes
1 answer
318 views

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 vote
1 answer
132 views

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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4 votes
0 answers
187 views

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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4 votes
1 answer
195 views

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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  • 255
0 votes
1 answer
386 views

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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  • 9
5 votes
1 answer
171 views

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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2 votes
2 answers
1k views

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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  • 131
1 vote
0 answers
161 views

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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1 vote
2 answers
256 views

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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6 votes
0 answers
78 views

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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  • 3,698
0 votes
1 answer
204 views

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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4 votes
1 answer
335 views

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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0 votes
1 answer
186 views

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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  • 119
4 votes
0 answers
632 views

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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  • 175
0 votes
0 answers
96 views

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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  • 608
0 votes
0 answers
90 views

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 ...
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  • 2,214
5 votes
0 answers
430 views

How to obtain the probability distribution of a sum of dependent discrete random variables more efficiently

I hope you are well. Here is my problem. Let $\{s_0,\,s_1,\ldots,\,s_T\}$ be a sequence of discrete random variables and denote $S_t=s_0+s_1+\cdots+s_t$, with $S_0=0$ and $S_T\leq M$, where $M$ and $T$...
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4 votes
0 answers
457 views

Sum of Binomial random variable CDF

Suppose there are two independent Binomial random variables $$ X\sim Binomial(n,p)\\ Y\sim Binomial(n,p+\delta) $$ where $\delta$ is considered to be fixed, and $p$ can vary in $(0,1-\delta)$. Now ...
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  • 103
2 votes
2 answers
885 views

Lower bound on number of samples for an epsilon delta approximation matching the Chernoff bound

So we have two biased coins, one comes out head w.p. $1/2+\epsilon$ and the other w.p. $1/2-\epsilon$. How many times should we flip these two coins to be able to tell them apart w.p. at least $\delta$...
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0 votes
1 answer
415 views

Tail bound for maximum of independent (but not identical) binomial random variables

This post derived the tail bound for the maximum of independent and identically distributed binomial r.v.'s based on normal approximation. Is there a similar result in the literature for finding the ...
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