All Questions
26
questions
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54
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Why is the multivariate normal distribution is $(\Sigma, C)$ sub-gaussian?
The definition of \textit{sub-gaussian} from a book I work with is: $X\in\mathbb{R}^n$ is $(\Sigma,C)$ sub-gaussian if $$\mathbb{P}(\lvert X^\top u\rvert>t)<Ce^{-t^2/(2u^\top\Sigma u)}, \qquad u\...
- 11
0
votes
1
answer
58
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Bounds on the ratio between second raw moment and expected of absolute value squared
I'm interested in bounds for the ratio $E[X^2]/E[|X|]^2$. The best lower bound is $1$ since $E[|X|] \geq E[X]$ and $E[X^2] - E[X]^2 = Var(X) \geq 0$. On the other hand, I would like to know if there ...
- 63
0
votes
0
answers
34
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2-Stirling number simplification with polynomial
I am looking for a way to either 1) simplify the following equation or 2) provide a reasonably tight upper bound to the following. Note that $\epsilon < 1$, and reasonably also $\epsilon \ll 1$.
\...
1
vote
0
answers
17
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Calculation of lower range confidence interval
Question
We observe $x$, the maximum of $n$ values in a random sample from the uniform distribution between $0$ and $c$, where $c > 0$. Find an exact lower range $100(1 - \alpha)\%$ confidence ...
- 2,187
0
votes
1
answer
58
views
Bound on the expected time of first success in a series of Bernoulli RVs
Given an infinite series of Bernoulli RVs $X_1,X_2,...$ (which may be differently distributed and mutually dependent), we are given that for every $n>0$, $$\mathbb{E}\left[\sum_{t=1}^{n}(1-X_t)\...
- 47
3
votes
1
answer
94
views
Minimum number of Bernoulli trials until sum reaches threshold with high probability
Let $X_1, X_2, \dots$ be i.i.d. $Bern(p)$ with $p\in (0, 1)$. Let $\delta \in (0,1)$ and $m \in \mathbb{N}$. What is the smallest integer $n \in \mathbb{N}$ such that $$P\left( \sum_{i=1}^n X_i \geq m ...
- 153
1
vote
0
answers
48
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Product of random variables greater than dependent sum of random variables
Let $X := {1\dots m}$ be the set of indexes corresponding to the elements of the vector of i.i.d. random variables $w:=(w_1, \dots, w_m)$. Let there be the subsets $Y, Z \subseteq X$ which may or may ...
- 221
1
vote
0
answers
27
views
Tight bounds for the expected maximum value of k IID Binomial(n, p) random variables
What is the tightest lower and upper bound for the expected maximum value of k IID Binomial(n, p) random variables
I tried to derive it :
$$Pr[max \leq C] = (\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^...
- 21
1
vote
0
answers
90
views
Proving the set where probability density function becomes infinite is bounded
For a continuous random variable $X$, with probability density function $p_X(x)$, it is known that there exists a $p_{min} > 0$ such that $p_X(x) \geq p_{min} \forall x \in X$. Also, I know that $X$...
- 11
1
vote
0
answers
48
views
Upper Bound for Moments for Product of Sample Means
I have a question about the upper bound of the following moment.
Suppose that $(A_1, e_1),\ldots, (A_n, e_n)$ are i.i.d. with $E(e_i)=0$. I am wondering if we have the bound
$$E\bigg(\bigg\|\frac{\...
- 53
0
votes
1
answer
31
views
Lower bound on empirical Gaussian probablity
Consider $\{a_1,a_2,...,a_n\}$ n points in $\mathbb{R}$. Assume their mean is $0$, their standard deviation is then given by $\sigma=\sqrt{\dfrac{1}{n}\sum_ia_i^2}$. Let $p(x)=\dfrac{1}{\sqrt{2\pi}\...
- 700
1
vote
1
answer
94
views
Probabilistic Bound on Random Walk with Drift
For Gaussian random variable $\xi_t$ with mean $\mu$ and standard deviation $\sigma$, consider the random walk with initial condition $P_0=100$, such that
\begin{equation}
P_t=P_{t-1}(1+\xi_t).
\end{...
- 175
0
votes
0
answers
60
views
Asymptotic propagation of error
Let $\tilde{s}_n$ and $\tilde{p}_n$ be estimators of the quantities $s$ and $p$, respectively ($\mathbb{E}[\tilde{s}_n]=s$ and $\mathbb{E}[\tilde{p}_n]=p$). Imagine we have obtained asymptotic bounds ...
- 984
1
vote
1
answer
47
views
Variance of sum of deviations
Suppose I have an i.i.d. sample $\{X_i\}_{i=1}^M$ for some positive integer $M$, and suppose that $X_i \sim X$ for some random variable $X$ with finite variance. Then, denote by
$$
E_M = \frac1M\sum_{...
- 1,450
0
votes
1
answer
110
views
Markov Inequality question confusion bound problem
"If a fair die is rolled 200 times, count the number of 1’s. Give an upper bound for the probability that the count of 1’s stays below 8. "
So here for Markov inequality,
P(X>=8)<=E[X]/8.
So here,...
- 1