All Questions
24 questions
2
votes
0
answers
76
views
Inequalities concerning cummulative distributions of binomials
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 $(m,p)$...
- 121
0
votes
0
answers
159
views
How to express the expectation and variance of a truncated binomial distribution without summation?
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 ...
- 1
3
votes
1
answer
195
views
Probability of sum of i.i.d. random variables being positive
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 ...
- 63
3
votes
1
answer
346
views
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} \...
- 85
0
votes
1
answer
159
views
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
$$P[X_{jn}= 1 ] = p_n = 1- P[X_{...
- 135
5
votes
3
answers
665
views
The relative error of approximating a binomial
Are there any good approximations for a binomial CDF that work well in terms of the relative error, as opposed to absolute? For the usual normal approximation, the absolute error is very well-studied ...
- 4,049
4
votes
0
answers
131
views
Log of a truncated binomial
Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
- 4,049
1
vote
1
answer
879
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)...
- 13
1
vote
1
answer
109
views
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 $x \in F_2^k$...
- 111
1
vote
1
answer
417
views
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 \...
user325283
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
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
- 331
-2
votes
1
answer
84
views
Ensemble averaging in a random graph (or network) in the large $N$ limit [closed]
I have a random graph/network described by the adjacency matrix $(a_{ij})_{N\times N}$ where $a_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta_i=\...
- 137
1
vote
1
answer
3k
views
Tail bound regime for Binomial distribution in concentration paper
In paper 'Concentration Inequalities and Martingale Inequalities:A Survey' gives the following inequality:
My question is whether the inequality holds in regime $\lambda$ being $o(\sqrt n)$ (say $\...
- 1,826
5
votes
0
answers
205
views
Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
- 101
4
votes
1
answer
272
views
How far do I have to go for the tail of a binomial distribution with small $p$ to be $O(1/n)$?
Let $n$ be a large integer, $p$ be a small number (say, $p=C/n$ for some constant $C \ll n$), and consider the tail of the binomial distribution $B(n,p)$, after $T$:
$$
\delta = \sum_{s=T}^{n} p^s (1-...
- 267
0
votes
1
answer
270
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 ...
- 173
4
votes
1
answer
648
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
213
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 : $$\forall 0 < t < 0.9, \quad \exists C, \quad \forall n >1, \quad \mathbb P\bigg(|X-np| \leq C\sqrt{n}\bigg) \...
- 119
4
votes
0
answers
823
views
Total Variation distance of polynomials of Bernoulli R.V.s
Let $X_i, Y_i$ be i.i.d Bernoulli $0/1$ random variables with
$\mathbb{E}[X_i] = p$ and $\mathbb{E}[Y_i] = q$.
Let
\begin{align*}
X &= X_1 X_2 + Χ_2 Χ_3 + \ldots +X_{n-2} X_{n-1}+ X_{n-1} X_n\\...
- 175
0
votes
0
answers
102
views
Lower bound for the probability that $X=\omega\left(\mathbb E[X]\right)$ for $X\sim Bin(n,p)$
Let $X\sim Bin(n,p)$ be a binomial variable and let $\delta\in (0,1)$.
I'm looking for a lower bound of the form $\Pr[X > f(\delta)] \ge \delta$.
Specifically, if $\delta,p=o(1)$ are not ...
- 618
5
votes
0
answers
523
views
How to obtain the probability distribution of a sum of dependent discrete random variables more efficiently
I hope you are well. Here is my problem.
Let $\{s_0,\,s_1,\ldots,\,s_T\}$ be a sequence of discrete random variables and denote $S_t=s_0+s_1+\cdots+s_t$, with $S_0=0$ and $S_T\leq M$, where $M$ and $T$...
- 173
4
votes
0
answers
516
views
Sum of Binomial random variable CDF
Suppose there are two independent Binomial random variables
$$
X\sim Binomial(n,p)\\
Y\sim Binomial(n,p+\delta)
$$
where $\delta$ is considered to be fixed, and $p$ can vary in $(0,1-\delta)$.
Now ...
- 103
2
votes
1
answer
190
views
Median of a uniform multinomial variable
Let $k\in\mathbb N^+$ be a positive integer.
Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$.
For $i\in \{1,2,\...
- 618
1
vote
0
answers
126
views
Quotient of cumulative binomial distribution functions
Given to integers $n < m \in \mathbb{N}_0$ and a probability $p$, I'm struggling to calculate (or at least get an upper bound for) the quotient
$$Q = \frac{F(n+1;m,p)}{F(n;m,p)}$$
where $F$ denotes ...
- 11