All Questions
11
questions
4
votes
1
answer
372
views
Exponential bound for tail of standard normal distributed random variable
Let $X\sim N(0,1)$ and $a\geq 0$. I have to show that $$\mathbb{P}(X\geq a)\leq\frac{\exp(\frac{-a^2}{2})}{1+a}$$
I have no problem showing that $\mathbb{P}(X\geq a)\leq \frac{\exp(\frac{-a^2}{2})}{a\...
- 103
2
votes
0
answers
92
views
Probabilistic bound on difference of Lipschitz random function
I am currently facing the following problem :
Let $(X_1,Z_1),\ldots,(X_n,Z_n)$ be $n$ i.i.d. sample points from some distribution $p$ supported on $\mathcal X\times\{-1,1\}$ where $\mathcal X\subseteq ...
0
votes
3
answers
230
views
Lower bound for $\sqrt{x} - \sqrt{x-1} $ [closed]
Does anyone know any useful lower bound for $\sqrt{x} - \sqrt{x-1} $ for $x>1$. I have a problem where I want to find a lower bound for $$\sqrt{C \log(n)} - \sqrt{C \log(n)-1} $$ for a positive ...
- 3
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
$$E\bigg(\bigg\|\frac{\...
- 53
1
vote
0
answers
52
views
Does logistic regression not fulfill an inequality required for Wilks' Theorem or am I missing something?
The required inequality:
Wilks' Theorem is given in the source below as Theorem 12.4.2, p. 515. Before stating the inequality, some definitions are needed:
Let $Z_1, \dots, Z_n$ be i.i.d. according to ...
- 47
2
votes
0
answers
212
views
Hoeffding's Inequality Assumptions
I'm looking for the assumptions of the Hoeffding's inequality to check it is applicable to my problem. So far the only assumptions I can find are the variables $Z_i$ are IID and bounded. However, Im ...
- 21
0
votes
1
answer
110
views
Markov Inequality question confusion bound problem
"If a fair die is rolled 200 times, count the number of 1’s. Give an upper bound for the probability that the count of 1’s stays below 8. "
So here for Markov inequality,
P(X>=8)<=E[X]/8.
So here,...
- 1
-1
votes
1
answer
3k
views
How to find upper bound of a probability?
We roll a fair die 50 times and count the number of 2’s. Give an upper bound for the probability that the count of 2’s stays below 7.
How can I approach this problem?
- 1
1
vote
0
answers
41
views
Nice bounds for 3rd folded central moment in terms of variance?
Say we have some real numbers $x_1, \ldots, x_n$, and let $\mu$ be their mean and $\sigma^2$ be the variance. Are there some nice bounds for $\frac 1n \sum_{i=1}^n |x_i - \mu|^3$ based on $\sigma^2$? ...
- 8,945
3
votes
1
answer
885
views
Lower bounds on sum of squared sub-gaussians
Letting $\left\{X_{i}\right\}_{i=1}^{n}$ be an i.i.d. sequence of zero-mean sub-Gaussian variables with parameter $\sigma,$ define
$Z_{n} :=\frac{1}{n} \sum_{i=1}^{n} X_{i}^{2} .$ Prove that
$$
\...
- 73
1
vote
1
answer
118
views
Inequality involving log-sum-exp, variance, and mean
Fix $z_1,\ldots,z_n \in \mathbb R$. Let $\mu_n:= mean(z_1,\ldots,z_n):=\frac{1}{n}\sum_{i=1}^n z_i$, $lse_n(z_1,\ldots,z_n)=\log(\sum_{i=1}^n e^{z_i})$, and
$\sigma^2_n := variance(z_1,\ldots,z_n):=\...
- 9,575