Questions tagged [bernoulli-numbers]

Questions on Bernoulli numbers, a special sequence of rational numbers that arise as the coefficients in the power series expansions of certain elementary functions.

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How to understand if an exercise is Poisson-Bernoulli or e -What are the difference

I have 3 different exercises and I am getting confused.If it is Poison or Bernoulli or e. I have seen and I know each and read, but I am not so clever to get it.I mean i can't detect the exercise what ...
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Bernoulli processes and information entropy

) Given two iid Bernoulli processes B1 and B2. B1 is denoted by a sequence of random variables with Xi={0,1} and Pr{Xi=1} = 1/3 and B2 is denoted by a sequence of random variables with {Yi}={0, 1, 2} ...
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Verifying this limit of a sum

A lenghty limit, $$\lim_{ n \to \infty}-\sqrt{8}\cdot \frac{n!}{B_n}\sum_{j=0}^{n}\frac{(1-2^{1-j})(1-2^{1+j-n})B_{n-j}B_j}{4^j(n-j)!j!}=\pi$$ The limit seems to appraoches to $\pi$ but I am not ...
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Euler sum with Bernoulli numbers

In many sources, I find such equality: $$\dfrac{1}{n}\sum_{k=1}^n \binom n k B_kB_{n-k}+B_{n-1}=-B_n$$ where $B_1=-\dfrac{1}{2}$ $$$$However, there don't write how to get it. I think that it's ...
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1answer
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Recurrence for A000670

A000670 contain a formula by Martin Kochanski: Recurrence: $2a(n)=(a+1)^n$ where superscripts are converted to subscripts after binomial expansion - reminiscent of Bernoulli numbers $B_n=(B+1)^n$. ...
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1answer
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Identity of Bernoulli polynomials [duplicate]

I am trying to prove the following identity: $$B_n(1-x)=(-1)^nB_n(x)$$ I know that $$B_n(1-x)=\sum_{k=0}^n\binom n k B_n\cdot (1-x)^{n-k}$$ (I do not know how to write here exactly what I want. So I ...
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Sum with Bernoulli numbers

How to prove that: $$\sum_{k=0}^n \binom n k 2^k B_k = (2-2^n)B_n$$ In this sum, $B_n$ is the Bernoulli number with $B_1 = -\frac 1 2$. Thanks for your attention!
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nth root of Bernoulli numbers

I'm trying to prove that the supremum limit is equal to infinity: $\limsup_{n->\infty}\sqrt[n]{|B_n|}=\infty$ Where $B_n$ is defined via the series expansion: $f(z)=\frac{z}{e^{z}-1}=\sum_{n=0}^{\...
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Can we view Bernoulli distribution as mixture of delta distribution?

Can we view a Bernoulli distribution Ber(p) as a mixture of delta distribution $p(x) = (1-p) \times \delta(x) + p \times \delta(x-1)$ Bernoulli distribution has ...
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2answers
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Computing CDF from an exponential r.v. X and Bernoulli r.v. Z

I need to compute the CDF of Y=ZX and i am struggling to compute this when Y=ZX Information: X is an exponential random variable with parameter 1 Z a Bernouilli random variable taking its values ...
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Is there a product formula for the Bernouilli numbers?

There are many well-known formulas available for the Bernouilli numbers, many of these can be found on the Wikipedia page. However, these are all in the form of infinite sums. I have not found any of ...
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Closed form for $\sum_{j=0}^{n}a^{-j}B_{j}B_{n-j}{n \choose n-j}$ [closed]

$n=2k+1$, $k\ge1$ Where $B_n$ ; Bernoulli number $$\sum_{j=0}^{n}2^{-j}B_{j}B_{n-j}{n \choose n-j}=-\frac{2^{n-2}+1}{2^n}\cdot nB_{n-1}\tag1$$ We manage to figure the closed form for $(1)$ We are ...
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Bernoulli numbers and $\pi^2$.

It is probably well-known that: $$ \lim_{n\to\infty}\frac{b_{2n}n^2}{b_{2n+2}}=-\pi^2, $$ where $b_n$ are the Bernoulli numbers. By a numerical experiment I have found that the quotient $$ \frac{b_{...
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For $p$ a prime larger than $2k+2$, is it possible that $p^2$ divides $B_{2k}$?

One characterization of the irregular primes is as follows: For an irregular prime $p$, there exists some natural number $k\le \frac{p-3}{2}$ such that $p$ divides [the numerator of] $B_{2k}$ where $...
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Probability of drawing a total of at least x amount of black balls from different boxes?

Suppose we have two boxes containing a 100 balls. Box A has 45 black, 55 white balls and Box B has 30 black and 70 white balls. There is no trick with the boxes and the drawing process is completely ...
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1answer
79 views

Rao Blackwell theorem on Bernoulli distribution

I am currently doing statistics homework and we just covered Rao blackwell theorem. The homework has 3 parts $X_1, \dots , X_n \sim \operatorname{Bernoulli}(p)$ : Give an unbiased estimator Give a ...
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Does $\sum\limits_{k=2}^{\infty}{\frac{|B_{k}|}{k!}(\cos(n)-1)}$ have a closed form?

I am trying to find a closed form expression of the following sum in terms of $n$ (if it exists) where $B_{k}$ is the $k$th Bernoulli number. $$\sum_{k=2}^{\infty}{\frac{|{B_{k}|}}{k!}(\cos(n)-1)}$$ ...
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About Bernoulli polynomials

My question is about Bernoulli numbers and Bernoulli polynomials in the $p$-adic context. In general in fact Bernoulli numbers are defined as global object so they do not depend on $p$. If $B_k(x)$ is ...
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Bernoulli numbers in the general series-expansion formula for sums of powers?

We know the series-expansion for these "sum of powers": $$\sum_{i=0}^n i = \frac{n(n+1)}{2} = \frac{1}{2}n^2+\frac{1}{2}n$$ $$\sum_{i=0}^n i^2 = \frac{n(n+1)(n+2)}{6} = \frac{1}{3}n^3+\frac{1}{2}n^2+\...
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2answers
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Recursive formula for Bernoulli numbers from power series

This is Ch 7, Exercise 62 in Palka's Complex Function Theory. Define $$ f:B(0,2 \pi) \to \mathbb{C},z \mapsto \frac{z}{e^z-1}\text{ if }z \neq 0, \; f(0):=1$$ Let $f$ have Taylor series $$f(z)= \sum_{...
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An identity involving binomial coefficients and Bernoulli numbers.

By solving a problem I have realized the following identity, which holds by numerical evidence: $$ \sum_{k=1}^i\frac1k\binom{i}{k-1}\binom kj{B_{k-j}}=\delta_{ij}. $$ where $B$ are the Bernoulli ...
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Show that X (not conditioned on p) is a Bernoulli random variable [closed]

Problem Statement: Suppose that $p ∼ U[0, 1)$. Show that $X$ (not conditioned on the value of $p$), is a Bernoulli random variable with pmf $ P(x) = \left\{ \begin{array}{lr} E(p) , &...
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Correlated Bernoulli Trials

Suppose there are $n$ dependent Bernoulli trials, $X _ { 1 } , \ldots , X _ { n }$ with $X _ { j } \in \{ 1,0 \}$ and $\operatorname { Pr } \left( X _ { j } = \right.$ 1) $= p$ for all $j = 1 , \...
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1answer
59 views

The denominator of a Bernoulli number is always **an even** integer. Why?

Apparently, the denominator of a Bernoulli number is always an even integer. Where does this come from?
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$p$-adic-valuation of an expression involving Bernoulli numbers

Let $p = 43, 67$ or $163$ (three primes such that $h(\mathbb{Q}(\sqrt{-p})) = 1$) and consider $k = (p+1)/2$. I'm interested in computing the $p$-adic valuation of the expression \begin{equation} 1+\...
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Prove 0<=entropy<=1 [closed]

The entropy of a Bernoulli random variable X with $P(X=1)=q$ is given by B(q)=-qlog(q)-(1-q)log(1-q) How do we prove 0<=B(q)<=1?
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Euler-Maclaurin Formula Definition Confusion

I am confused about these $2$ definitions of the Euler-Maclaurin formula. I read the following here: The full Euler-Maclaurin formula with no remainder term (for infinitely differentiable $f$ ) ...
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Complex Analysis Extension of Bernoulli Number Generating function

I have just recently started revising for my complex analysis module at university and come across and interesting exercise in a textbook while reading. I vaguely understand the concept but am ...
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Proving Chebyshev inequality

Let $\xi_1,...,\xi_n$ be independent and equally distributed Bernoulli random variables. Also let $\mathbb{P}(\xi_i=1)=p,$ $\mathbb{P}(\xi_i=-1)=1-p$ where $i=1,...,n.$ Prove that $$\mathbb{P} \Bigg(...
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Proving $\mathbb{P}(\xi_1+ \xi_2+…+\xi_n=1)=(\sum_{i=1}^{n}\lambda_i)\Delta + \mathcal{R}\Delta^2$

Let $\xi_1, \xi_2,...,\xi_n$ be independent Bernoulli random variables in $(\Omega,\mathcal{P}(\Omega),\mathbb{P})$ and $$\mathbb{P}(\xi_i=0)=1-\lambda_i\Delta$$ and $$\mathbb{P}(\xi_i=1)=\lambda_i\...
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Maximizing the length of the confidence interval

Let $X_1,X_2,...,X_n$ be iid $Bern(θ)$. (a) Determine the sample size such that the length of 95% confidence interval is at most 0.04. (b) To reduce the cost, assume that the half of the sample is ...
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Is the series $‎\sum_{n=0}^{‎\infty‎}‎\frac{B_n(z)}{2^n}‎$ convergent?

‎$B_n(z)‎‎$‎‎ ‎is the Bernoulli polynomial, ‎we know that the series $$\sum_{n=0}^{‎\infty‎}‎\frac{B_n(z)}{n!}‎$$ ‎is ‎convergent ‎for ‎every ‎‎$z\in‎\mathbb{C}‎$‎ ‎and its‎ ‎closed form ‎is equal ...
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Cumulants of discrete uniform distribution from Bernoulli numbers

Let $X$ be the random variable uniformly distributed on $\{0,\dots,d-1\}$. This is, $$\operatorname{Prob}(X = i) = \frac{1}{d}$$ for $i \in \{0,\dots,d-1\}$. Computations suggest that the cumulants of ...
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1answer
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Bernoulli's inequality and sequences

How can I use Bernoulli's inequality to prove $c^{1/n}\to1$ for $c>0$ ? I have to use $c^{1/n}=1+xn$. Thank you!
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1answer
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Function to calculate t-stat of similarity in survey answers

I have a survey of a large number of questions. Each question is multiple choice, and has three possible answers. Users get served random questions to answer. So they do not all answer the same ...
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What value of Y have a Bernoulli (1/2) distribution?

Let $X$ be a random variable with p.d.f. $$f_X(x) = \frac{1}{\beta}e^{-x/\beta} 1_{[0,∞)}(x)$$ , where $\beta > 0$. For a positive number $b$, let us define a random variable $Y$ as $$Y =\...
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1answer
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Find linear recurrence for set constructed by Bernoulli trial

Got such a task at the exam. Any ideas how to solve it? Consider the set $ \{1,\dots,n\} $. Lets perform sequential pairwise independent Bernoulli trials with success probability $ \frac{1}{2}$. If ...
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1answer
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Bernoulli Process, how to define as independent trials

Consider a Bernoulli Process(p). What is the probability you will get 5 heads before 3 tails, flipping a coin. Im having trouble answering this question because, from my understanding Bernoulli ...
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I need some advice on how to organize some drafts and what to with a new formula I found

Found a new formula to get the sum of k to the power of any positive integer , k from 1 to n .I know bernoulli already found another formula in which he created his bernoulli numbers. This new fomula ...
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Very small constants from poly-bernoulli

If we define $$a_{n}(m)=\sum\limits_{k=0}^{n}k!{n\brace k}(k+1)^m(-1)^{n-k}$$ $$\prod\limits_{k=2}^{n+1}1-kx=\sum\limits_{k=0}^{n}t(n,k)x^k$$ for $n>0$, $m\geqslant0$, so $$\sum\limits_{k=0}^{n}t(n,...
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Closed form expression for $\sum_{k=0}^\infty \frac{{\rm B}_{k+1}}{(k+1)!} \, \frac{(k-s)!}{k!(-s)!} \, \left(-x\right)^k$

I'm wondering if there is a closed form expression for $$ \sum_{k=0}^\infty \frac{{\rm B}_{k+1}}{(k+1)!} \, \frac{(k-s)!}{k!(-s)!} \, \left(-x\right)^k $$ where ${\rm B}_k$ are the Bernoulli numbers ...
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1answer
124 views

How to calculate a Bernoulli Distribution problem

I have my statistics exam quite soon and i came upon this question : At the last referendum, $40\%$ of the Italian population supported the constitutional reform. If a random sample of size $n = ...
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Closed expressions for divergent series over Bernoulli numbers?

Motivation In a recent post (Asymptotic behaviour of sums involving $k$, $\log(k)$ and $H_{k}$) I asked for the asymptotic behaviour of the sum $$\sigma_{c}(n)=\sum_{k=1}^n H_{k} \log(k)\tag{1}$$ ...
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57 views

Does the sum $\sum_{k=1}^{\infty}B_{(4k-2)}+B_{(4k)}$ converges?

Just as the title says I'd like to know if this sum $$ \sum_{k=1}^{\infty}(B_{(4k-2)}+B_{(4k)}) $$ converges and if so to which value. Here $B_{2k}$ are Bernoulli numbers. I've tried with Mathematica ...
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1answer
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How do I construct Faulhaber's Triangle?

There's a youtube video on the construction, but the chalk board is blurry, and I've seen 3 different versions, including Jacob Bernoulli's (shown in wikipedia/faulhaber's formula) which omits the ...
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1answer
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Notation: What does “$p-1|n$” mean in “$\prod_{p-1|n} p$”?

I'll try my best to reconstruct how this appears in my pdf: $$d:denom(B_n)=\prod_{p-1|n} p$$ the "$p-1|n$" part, I don't understand. I get this has something to do with Riemann Zeta Function, and ...
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On the Puiseux series of divergent zeta function for $0 < \Re(s)< 1$

Let $s$ be a complex number such that 0 < $\Re(s) < 1$ and $\zeta(s)$ be the analytic continuation of zeta function on the strip 0 < $\Re(s) < 1$. Then by applying the Euler-Maclaurin ...
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45 views

Ordinary Bell polynomials of Bernoulli Numbers.

I am working on a problem that involves this sum: $$\sum_{k=1}^n B_{n,k}^o \left(-\frac{B_2}{2c}, -\frac{B_4}{4c}, -\frac{B_6}{6c}, \cdots, -\frac{B_{2(n-k+1)}}{2(n-k+1)c}\right)$$ We need to ...
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2answers
41 views

Bernoulli Number Sum: At B4 i get 1/180?

I know i did something wrong because i don't get -1/30. I was able to get zero for B3 and 1/6 for B2, but then $$B_4=1-{4\choose 0}\frac{B_0}{4-0+1}-{4\choose 1}\frac{B_1}{4-1+1}-{4\choose 2}\frac{...
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3answers
447 views

What is the simplest way to get Bernoulli Numbers?

on paper, long hand, what is the simplest way to generate the bernoulli fractions like -1/30 and 7/6? Basically im trying to find and understand Bn = (the stuff on this side) and ive seen something ...