Skip to main content

All Questions

Filter by
Sorted by
Tagged with
3 votes
1 answer
345 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} \...
reservoir's user avatar
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\...
Tom Solberg's user avatar
  • 4,049
3 votes
0 answers
95 views

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 ...
Argemione's user avatar
  • 131
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}\...
香结丁's user avatar
  • 331
0 votes
1 answer
314 views

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}\...
qwert's user avatar
  • 89
0 votes
0 answers
106 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 ...
abs135's user avatar
  • 1
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 ...
R B's user avatar
  • 618