Questions tagged [binomial-distribution]

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1answer
36 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=\...
-2
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1answer
98 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^...
1
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2answers
94 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 ...
2
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1answer
138 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 ...
1
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0answers
136 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 ...
0
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1answer
123 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 $\...
1
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1answer
113 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 ...
4
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0answers
177 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 ...
4
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1answer
176 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-...
0
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1answer
108 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: ...
5
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1answer
150 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 ...
1
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2answers
246 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?
1
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0answers
151 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\...
1
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2answers
243 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 ...
6
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0answers
73 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 ...
0
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1answer
110 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 ...
4
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1answer
179 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 $X$ ...
0
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1answer
161 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) \...
5
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0answers
369 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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0answers
95 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 ...
0
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0answers
88 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 ...
5
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0answers
368 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 $T$...
4
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0answers
405 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 ...
2
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2answers
557 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 $\delta$...
0
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1answer
293 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 ...
3
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2answers
1k views

cumulative binomial equals integral

Can somebody explain why the cumulative binomial equals an integral expression? Thanks!${}$ $$ \sum_{j=0}^{k-1}\binom{n}{j}\theta^j(1-\theta)^{n-j} = 1- \binom{n}{k}k\int_0^\theta t^{k-1}(1-t)^{n-k}...
2
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1answer
135 views

Median of a uniform multinomial variable

Let $k\in\mathbb N^+$ be a positive integer. Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$. For $i\in \{1,2,\...
4
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1answer
2k views

Fundamental difference between Poisson Point Process and Binomial Point Process

What is the fundamental difference between Poisson Point Process and Binomial Point Process? I am evaluating a solution in a Binomial Point Process setup. If I want to evaluate that in a Poisson ...
0
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1answer
132 views

Proving an inequation based on binomial distributions

Problem statement Let $c \in \mathbb{N}$, $n_1 \in \mathbb{N}_0$, and $n_2 \in \mathbb{N}_0$ be integers and $p$ a probability. Furthermore, let $b(m,j,p) = \binom{m}{j}p^j(1-p)^{m-j}$ denote the ...
1
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0answers
71 views

Importance sampling for bernoulli-sequence, favouring long sequences of ones

Assume we have a sequence of i.i.d. bernoulli-distributed random variables of length $n$. I'm interested in doing rare event simulation and my event depends, among other random factors, on the ...
1
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0answers
95 views

Quotient of cumulative binomial distribution functions

Given to integers $n < m \in \mathbb{N}_0$ and a probability $p$, I'm struggling to calculate (or at least get an upper bound for) the quotient $$Q = \frac{F(n+1;m,p)}{F(n;m,p)}$$ where $F$ denotes ...
4
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1answer
1k views

variance of compound binomial distributions

The below is motivated by a problem I'm observing in my experimental data I have m boxes, where each box is supposed to contain k molecules of mRNA. The measurement process includes labeling all the ...
9
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2answers
976 views

Expected centered entropy of the binomial distribution

In short, the function I am interested in is the following: $$I_n(p) = \sum_{k=0}^n \binom{n}{k} p^k (1-p)^{n-k} \left[h(p) - h\left(\tfrac{k}{n}\right)\right],$$ where $h(x) \triangleq -x \log x - (1-...
6
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0answers
656 views

Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson

Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites. Let the ...
10
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1answer
706 views

Binomial distribution conjecture

Conjecture: Let $m$ and $n$ be fixed positive integers and let $f(k)$ be the probability that a Binomial($k(m+n)$, $p$) random variable is less than $kn$. Then for sufficiently small $p$, $f(k)$ is an ...
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0answers
838 views

Markov transition probabilities and negative binomial distribution.

A realization of a Markov process generates a sequence of interval lengths between transition from one state to another. A natural way of modeling the distribution of the lengths is as a negative ...
6
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1answer
622 views

Calculating a specific joint probability involving sums of binomial distributions

The following might look like a simple problem - but the question has been unanswered for more than a week on math.stackexchange.com, and I have asked quite a few of the Ph.d. students at our ...
1
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1answer
903 views

central limit theorem for binomial random variable

I'm confused about applying central limit theorem to Bernoulli random variables. Let $X_i=\frac{n}{\sqrt{n-1}}(Z_i - \frac{1}{n})$ where $Z_i$ is iid Bernoulli($\frac{1}{n}$). Then $E[X_i]=0$ and $...
0
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1answer
307 views

Analytical expression for variance of nested binomials?

Hi all, I want to compute the variance of a variable that is defined at each step as a recursion of binomials in the following way: A=1 B=Bin(1,A)*Bin(1,p) C=Bin(1,B)*Bin(1,p) D=Bin(1,C)*Bin(1,p),...
18
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7answers
9k views

Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer. Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case $\...
2
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1answer
1k views

Probability of system failure in a distributed network

I am trying to build a mathematical model of the availability of a file in a distributed file-system. The system works like this: a node $x$ stores a file $f$ (encoed using erasure codes) at $rb$ ...
2
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1answer
2k views

Generalizing the wilson score confidence interval to other distributions

This article describes the 'Wilson score confidence interval', and describes how to use it to derive the lower bound on the nth percentile confidence interval for determining sorting criteria for ...
4
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3answers
4k views

Range of binomial probability, given a certain number of observations?

Let's say I am given $n$ flips of a coin, $k$ of which are heads. These are iid flips. Can I say, with probability $p > 1/2$, that the true probability of heads is in range $[p_1, p_2]$ ? What is ...
2
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3answers
595 views

When do binomial distributions occur?

A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not ...