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Questions tagged [binomial-distribution]

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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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Range of convergence for 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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Binomial CDF - Wiki vs Reality [migrated]

In https://en.wikipedia.org/wiki/Binomial_distribution the binomial CDF is defined as $F_{Bin(n,p)}(k) = I_{1-p}(n-k, k+1)$, Where $I$ is the regularized incomplete beta function. There is a proof ...
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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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Concavity of a function after binomial transform and maximization

Suppose $f(x)$ is a one-dimensional discrete concave function, and $X(n,p)$ is Binomial random variable. Is the function $g(x)= \max_n \mathbb{E} f(x+X(n,p))$ also discrete concave?
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How can I solve a constrained optimization problem with a random number of decision variables?

Here is my problem. Let $A_t$ be a random variable with Poisson-Binomial distribution with set of success-probabilities $\{q_1^{(t)},\ldots,\,q_n^{(t)}\}$, with $t\in\{1,\,2,\,3,\ldots\}$, $n\in\...
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1answer
63 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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1answer
160 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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1answer
138 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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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 ...
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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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294 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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2answers
295 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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1answer
163 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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2answers
966 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}...
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Using Binomial Distribution for analysis

Let $X\sim Bin(n,p)$ such that $p>\frac{1}{2}$. I know that $\forall k\in \{1,\ldots, n\}$ the following holds: $Pr(X\geq k)= \displaystyle{\sum_{t=k}^n} {{n}\choose{t}} p^t (1-p)^{n-t}$. But is ...
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1answer
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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,\...
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1answer
1k 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 ...
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1answer
124 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 ...
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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 ...
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86 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 ...
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1answer
922 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 ...
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2answers
830 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-...
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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 ...
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1answer
682 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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773 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 ...
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1answer
618 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 ...
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1answer
811 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 $...
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1answer
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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),...
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7answers
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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 $\...
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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 $rb$ ...
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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 ...
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3answers
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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 ...
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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 ...