All Questions
Tagged with lower-bounds probability-distributions
14 questions
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} \...
- 85
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 $$\frac{\boldsymbol{E}\log\...
- 4,049
2
votes
1
answer
1k
views
Finding the expectation $\mathrm{E} (1/ X)$ for a negative binomial random variable $X$
Suppose a random variable $X$ is distributed as $\operatorname{NB}(\mu, \theta)$, and its mass is as follows
$$ \mathrm{P}(X = y) = \binom{y + \theta - 1}{y} \left(\frac{\mu}{\mu + \theta}\right)^{y}\...
- 331
2
votes
1
answer
464
views
Lower-bound for $E[\min(X, k)]$ where $X$ is sum of Bernoulli random variables with $E[X]$ being a linear function of $k$
Given a real number $\alpha \in [0.5, 1.5]$, an integer number $k>1$, and a set of independent Bernoulli random variables $x_1, \dots, x_n$, I am interested to find a lower-bound for $F(\alpha, k)= ...
- 189
0
votes
1
answer
295
views
Are there known bounds on the expectation of the truncated Beta distribution?
Let $X\sim beta(\alpha,\beta)$ be a random variable and let $\tau\in(0,1)$.
Are there any known closed-form bounds (I'm specifically interested in lower bounds) on
$$
\mathbb E[X\ | X\le \tau]?
$$
- 618
2
votes
1
answer
280
views
Complicated bound after using Stirling's approximation
I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{...
- 225
3
votes
1
answer
165
views
Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$
Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...
- 6,853
3
votes
1
answer
167
views
Lower bound on the sum of pmf squared of a hypergeometric distribution
I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...
- 33
1
vote
1
answer
91
views
A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported
I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...
- 6,853
1
vote
0
answers
108
views
Bounding quantiles of the noncentral chi distribution
I need to bound the empirical quantiles for a noncentral chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality ...
- 162
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
4
votes
1
answer
232
views
Expectation over Pareto Sums
Given $K$ iid random variables $x_i$ with uniform distribution on $(0,1]$
and a constant $\alpha > 0$, the random variable $x_i^{-\alpha/2}$ is Pareto-distributed with scale parameter $1$ and ...
- 41
4
votes
1
answer
749
views
Lower bound on the tail of the hypergeometric distribution
Suppose there is a bag with $M$ white marbles and $N - M$ black marbles. Let $H(n, N, M)$ be a random variable which is number of white marbles in a draw, without replacement, of $n$ marbles from a ...
- 2,221
6
votes
2
answers
4k
views
tight bounds on probability of sum of laplace random variables.
Are there tight upper and lower bounds on the density of the sum of $n$ i.i.d laplace random variables that depend on $n$ and the individual laplacian densities?
- 111