Skip to main content
MathOverflow

Questions tagged [binomial-distribution]

Ask Question

The tag has no summary.

89 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
1 vote
1 answer
220 views

Let $X \sim \text{Bin}(n, p)$, and let $w\geq 0$ be some number such that $w \lll n$. For concreteness, say $p = 1/3$ and $w = \log^2 n$, for example. How can I bound the total variation distance ...
Jonathan Lu's user avatar
  • 11
5 votes
1 answer
391 views

I have a binomial probability question. I have run simulations up to $1,000,000$ and it seems to hold true, but the inequality seems quite delicate. Let $k,l$ be positive integers s.t. $k>l$ and ...
MDR's user avatar
  • 348
3 votes
1 answer
241 views

Is it possible to prove that a $Beta(\alpha,\beta)$ distributed random variable is always stochastically dominated by a $Beta(\alpha',\beta')$ if $\alpha'>\alpha$ and $\beta'<\beta$? I've read ...
Marco Max Fiandri's user avatar
  • 55
0 votes
1 answer
133 views

Statement Assume that $\sigma\in (1,+\infty)$, $N\in[3,+\infty)\cap \mathbb{Z}$, $p,t\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that $$[B(n_1)]^\frac{1}{\sigma} - [B(n_1+1)]^\frac{1}{\...
John_zyj's user avatar
  • 37
2 votes
0 answers
101 views

For random variable $Z$, let $F_Z$ denote its cdf, i.e., $F_Z(t)=\mathbb{P}(Z\leq t)$. Let $X$ be a binomial distribution with parameters $(n,p)$ and $Y$ a binomial distribution with parameters $(m,p)$...
Marcos Kiwi's user avatar
  • 121
0 votes
0 answers
240 views

Given a binomial distribution with parameters $ n $ and $ p $, where $ n $ is an odd integer greater than or equal to 3, I am interested in the truncated binomial distribution where we truncate at $ k ...
GodsDusk's user avatar
  • 1
3 votes
1 answer
260 views

Let $g,l \in (0,1)$ and $p\in [0,1]$. Let $X(k,1-p)$ be a random variable with binomial distribution with parameters $k$ and $1-p$. Let $Y(k,p)$ be a random variable with binomial distribution with ...
EGT's user avatar
  • 63
1 vote
1 answer
220 views

Statement Assume that $\sigma,R\in (1,+\infty)$, $N\in\mathbb{N}^*$, $p\in (0,1)$, $n_1\in\{0,1,2,\cdots,N-1\}$. Prove or disprove that $$B^\frac{1}{\sigma}(n_1)-B^\frac{1}{\sigma}(n_1+1)<1 .$$ ...
John_zyj's user avatar
  • 37
3 votes
1 answer
130 views

Let $X$ be a binomial random variable with parameters $n,p$. Define the function $f(n, p, t) = E\frac{1}{1 + t X}, $ where $t > 0$. Question: Can we find an elementary function $F(n, p, t)$ such ...
Drew Brady's user avatar
  • 552
2 votes
1 answer
103 views

This question is a twist on a question I asked here Random pseudo-walk of Poisson variables, but with randomly 'disappearing' objects. I do not know how to generalize the (satisfactory) answer given ...
Amir Ban's user avatar
  • 61
2 votes
1 answer
224 views

I would like to prove that the following function, for an arbitrary integer $n$: \begin{equation} \begin{split} f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\ & = x \cdot \sum_{k=0}^{n} \...
RotemBZ's user avatar
  • 23
2 votes
1 answer
299 views

Let: $F_{j+1,y}(s)$ be the cumulative distribution function of a binomial distribution with mean $y$, $j+1$ independent trials considered for $s$ successes. Is it possible to show in any way that: $\...
Marco Max Fiandri's user avatar
  • 55
3 votes
1 answer
534 views

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} \...
reservoir's user avatar
  • 85
0 votes
1 answer
109 views

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 ...
Štěpán Pardubický's user avatar
  • 1
0 votes
1 answer
213 views

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_{...
Fam's user avatar
  • 135
5 votes
3 answers
771 views

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 ...
Tom Solberg's user avatar
  • 4,131
4 votes
0 answers
155 views

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\...
Tom Solberg's user avatar
  • 4,131
1 vote
1 answer
1k views

(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)...
Renel's user avatar
  • 13
6 votes
3 answers
922 views

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 ...
Jerry's user avatar
  • 77
2 votes
1 answer
180 views

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 ...
jlewk's user avatar
  • 1,844
1 vote
1 answer
143 views

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 $x \in F_2^k$...
manmatha.roy's user avatar
  • 111
5 votes
1 answer
233 views

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}\...
Taras Banakh's user avatar
  • 44.4k
3 votes
0 answers
123 views

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 ...
Argemione's user avatar
  • 131
3 votes
0 answers
127 views

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 {...
Synia's user avatar
  • 593
1 vote
1 answer
112 views

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 ...
Tristan Nemoz's user avatar
  • 167
1 vote
1 answer
484 views

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 \...
user avatar
1 vote
1 answer
99 views

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 ...
kensk's user avatar
  • 13
0 votes
2 answers
560 views

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{...
qwert's user avatar
  • 73
4 votes
0 answers
437 views

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\...
neitherother's user avatar
  • 41
0 votes
1 answer
195 views

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} \...
Silvester's user avatar
  • 49
4 votes
1 answer
196 views

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 \...
Silvester's user avatar
  • 49
1 vote
1 answer
304 views

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 ...
messi22's user avatar
  • 53
1 vote
2 answers
216 views

If I have a binomial $X \sim B(n,p)$, and another binomial $X' \sim B(n,p)$ conditioned on $X'$ being of even parity. Is it true that there always exists a coupling for $(X,X')$ with $|X-X'| \le 1$? (...
DJA's user avatar
  • 445
2 votes
1 answer
1k views

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}\...
香结丁's user avatar
  • 341
0 votes
1 answer
392 views

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}\...
qwert's user avatar
  • 73
2 votes
1 answer
273 views

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\...
qwert's user avatar
  • 73
7 votes
0 answers
386 views

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 ...
nichehole's user avatar
  • 381
0 votes
0 answers
156 views

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 ...
abs135's user avatar
  • 1
-1 votes
1 answer
363 views

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 ...
Mateus Araújo's user avatar
  • 712
11 votes
2 answers
1k views

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 ...
Pat Devlin's user avatar
  • 2,752
3 votes
1 answer
300 views

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 ...
Pat Devlin's user avatar
  • 2,752
3 votes
2 answers
179 views

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}\!\! \...
user avatar
-2 votes
1 answer
115 views

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=\...
maurizio's user avatar
  • 137
-2 votes
1 answer
149 views

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^...
Andjela Todorovic's user avatar
  • 11
5 votes
1 answer
471 views

Given an integer number $m>0$ and a real number $\alpha\in [1, 2]$, I am interested in finding a lower-bound for $\Pr[X\geq m]$ subject to $X \sim \text{Binomial}(n, m\alpha/n)$. For large values ...
Melika's user avatar
  • 199
1 vote
2 answers
226 views

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 ...
VS.'s user avatar
  • 1,846
2 votes
1 answer
297 views

$\mu=1+\epsilon$ 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 interpreted ...
VS.'s user avatar
  • 1,846
1 vote
0 answers
241 views

$f_{n,p}(k)$ 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 ...
user151031's user avatar
  • 11
2 votes
1 answer
3k views

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 $\...
VS.'s user avatar
  • 1,846
1 vote
1 answer
198 views

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 ...
user2679290's user avatar
  • 340

1
2