Questions tagged [binomial-distribution]
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76
questions
2
votes
1
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Proving convexity of the expected logarithm of binomial distribution
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} \...
2
votes
1
answer
105
views
Convexity of a function
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:
$\...
2
votes
1
answer
247
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Simple anticoncentration bound for binomially distributed variable
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} \...
0
votes
1
answer
51
views
Expectation of beta function with binomial distribution
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 ...
0
votes
1
answer
118
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Approximation of a random sum of random variables (infinitely divisible distribution) by a triangular array
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
$$P[X_{jn}= 1 ] = p_n = 1- P[X_{...
5
votes
3
answers
629
views
The relative error of approximating a binomial
Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...
4
votes
0
answers
120
views
Log of a truncated binomial
Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
1
vote
1
answer
666
views
Does the (normalized) product of two independent binomial variables converge in distribution to a normal variable?
(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)...
6
votes
3
answers
848
views
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 ...
2
votes
1
answer
94
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 ...
1
vote
1
answer
105
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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 $x \in F_2^k$...
4
votes
1
answer
187
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}\...
3
votes
0
answers
90
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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 ...
3
votes
0
answers
77
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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 {...
1
vote
1
answer
77
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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 ...
1
vote
1
answer
330
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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 \...
1
vote
1
answer
61
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 ...
0
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2
answers
416
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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{...
4
votes
0
answers
405
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\...
0
votes
1
answer
149
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} \...
4
votes
1
answer
161
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 \...
1
vote
1
answer
230
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 ...
2
votes
1
answer
1k
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
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
0
votes
1
answer
291
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better lower (and upper) bound for $i$'s moment of function of binomial random variable with $i = \frac{1}{j}, j \in \mathbb{N}$
I want to derive a lower bound for $E\left[\left(\frac{X}{k-X}\right)^{i}\right] $ with $X \sim Bin_{(k-1),p}$ and $ k \in \mathbb{N} $. So far I could prove that
\begin{equation}
E\left[\frac{X}{k-X}\...
2
votes
1
answer
203
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sign of odd central moments of binomial distribution
I am interested in the sign of odd, central moments of a binomial distribution. From DOI. 10.1137/070700024 I have the formulae:
$ E\left[\left(X-\mu\right)^d\right]= \sum_{i=0}^n\binom{n}{i}\left(-p\...
7
votes
0
answers
306
views
Estimating the alternating sum $\sum_{j \ge 1} (-1)^j e^{-j^2} j^k$
I have been trying to get a lower bound on the following alternating sum but without much success:
$$
\sum_{j=1}^T (-1)^j e^{-j^2} j^k .
$$
For small values of $k$, this is easy because the first term ...
0
votes
0
answers
90
views
Upper bounding the sum with hypergeometric and binomial probabilities
Could you please help me upper bound this tricky expression:
$$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}$$.
So far I only ...
-1
votes
1
answer
237
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Lowerbounding expectation value of binomial tail
I'm trying to find a lower bound for the following expression for $q\ge p$:
$$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}.$$
It can be thought of as the ...
7
votes
1
answer
955
views
Bounding the probability that two binomials are equal
Note: This question was migrated from this earlier post, where it initially appeared. Following suggestions, I moved this into its own question.
Let $B_{n,p}$ denote the usual binomial random ...
3
votes
1
answer
212
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Inequality for difference of consecutive atom probabilities for binomial distribution
Edit: This post was originally two questions, the first of which has been answered, but a reference would still be appreciated if existent. The second question has been removed and migrated to its ...
3
votes
2
answers
150
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What is limiting behavior for the median of $2^X+2^Y$, where $X$ and $Y$ are iid binomial variables?
Suppose $X_n$ and $Y_n$ are iid variables, each distributed binomially as $B(n,1/2)$.
Let $M_n$ be the median of $2^{X_n}+2^{Y_n}$. Empirically
$$\sqrt{2}\, \le \left(\frac{M_n}{2}\right)^{1/n}\!\! \...
-2
votes
1
answer
64
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
votes
1
answer
111
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
vote
2
answers
142
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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
votes
1
answer
222
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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
vote
0
answers
220
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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 ...
1
vote
1
answer
2k
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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
vote
1
answer
142
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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
votes
0
answers
198
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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
votes
1
answer
235
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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 $T$:
$$
\delta = \sum_{s=T}^{n} p^s (1-...
0
votes
1
answer
611
views
The mean E(X) of negative binomial distribution [closed]
What I know about the mean of the negative binomial distribution is E(x)=r(1-p)/p. but there are some questions use E(x)=r/p as the mean. Very confusing and I don't understand at all.
For example:
...
6
votes
1
answer
198
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Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$
Disclaimer. Question moved from SE.
Setup
Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$.
Question
What is a good upper-bound for $\mathbb E[|X-np|^r]$ ?
Solution for small $r$
If $r=2$, then ...
4
votes
2
answers
2k
views
Can the sum of identically distributed dependent Bernoulli trials be binomially distributed?
If you have $n$ identically distributed Bernoulli trials whose sum is binomially distributed random variable, does it then follow that the $n$ Bernoulli trials are independent?
1
vote
0
answers
189
views
Rate of convergence for difference between conditional and marginal probability
Suppose $X\sim \text{Bin}(2n,p)$ and $X_1,X_2\sim\text{Bin}(n,p)$ are independent, with $X_1+X_2=X$. I'm interested in the rate of convergence for the absolute difference
$$
\left\vert P(X>c|X_1\...
1
vote
2
answers
278
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
votes
0
answers
82
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
votes
1
answer
263
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
votes
1
answer
572
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
votes
1
answer
209
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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) \...
4
votes
0
answers
767
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\\...