2
votes
0answers
33 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
1
vote
0answers
68 views

Concentration inequality of weighted sum of random variables given a tail inequality

I tried to solve the exercise below which can be seen as a generalization of the Bernstein's concentration inequality. However, I have difficulty bounding the moment generating function of $Z$ (see ...
1
vote
1answer
21 views

limiting behavior of standard normal survivor function [duplicate]

How do you show that $\lim_{x\to \infty} 1-\Phi(x) \sim \phi(x)/x$? In the previous, I'm using $\Phi$ to refer to the standard normal CDF and $\phi$ to refer to the standard normal pdf. Thanks!!
0
votes
0answers
6 views

Is the following a proper application of Hoeffding's inequality?

I'm somewhat shaky on my probability and was wondering if someone could double-check that the following is a valid application of Hoeffding's inequality. Suppose that I have a random variable $X \sim ...
0
votes
0answers
29 views

Proof also valid for discrete random variable?

In Theorem 2.2. in this paper you can find a proof of the one-sided Chebyshev inequality $$Pr[X \geq \mu +a ] \leq \frac{\sigma^2}{\sigma^2+a^2}$$ for a random variable $X$ with mean $\mu$ and ...
2
votes
1answer
98 views

Is Wikipedia correct?

Cantelli's Theorem Wikipedia says: $$P[X-\mu \geq a] \leq \frac{\sigma^2}{\sigma^2+a^2}$$ for $a > 0$ and $$P[X-\mu\geq a] \geq 1- \frac{\sigma^2}{\sigma^2+a^2}$$ for $a <0$. Is the second ...
1
vote
1answer
21 views

“Reverse” distribution-tails

Chernoff, Markov and Chebyhev all give some upper bound for tail probabilities, e.g. Chebyshev gives us $Pr[|X-E[X]| \geq t] \leq \frac{Var[X]}{t^2}$. This is quite helpful, but what if I would ...
1
vote
1answer
73 views

Tail bound sum of unbounded RVs of given mean and variance?

Let $X_1,\dots,X_n$ be independent variables, $X_i$ having mean $\mu_i$ and variance $\sigma_i^2$. Let their sum $S = \sum_{i=1}^n X_i$. Of course, $S$ has mean $\lambda = \sum_{i=1}^n \mu_i$ and ...
1
vote
1answer
220 views

difference between upper and lower tails of Chernoff bounds

I'm struggling with the intuition behind why Chernoff bounds differ in the upper and lower tails. That is, for the lower tail we have: $$ Pr(X \le (1 - \delta)\mu)\ \ \le\ \ ...
2
votes
1answer
155 views

'Simple" Chernoff bounds

Reading an academic paper I've observed the next claim: A simple Chernoff argument will now show that if an event has a constant probability at every step of occurring and there's independence ...
1
vote
2answers
196 views

Coins and probability

Bob has $n$ coins, each of which falls heads with the probability $p$. In the first round Bob tosses all coins, in the second round Bob tosses only those coins which fell heads in the first round. Let ...
0
votes
1answer
234 views

Upper-tail inequality for t-distribution

I am interested in upper tail bounds (or bounds on deviation from the mean) for t-distribution with n degrees of freedom (http://en.wikipedia.org/wiki/Student's_t-distribution) A bound that is of the ...
5
votes
2answers
378 views

Repeatedly rolling a die and the tails of the multinomial distribution.

For $1\leq i\leq n$ let $X_i$ be independent random variables, and let each $X_i$ be the uniform distribution on the set ${0,1,2,\dots,m}$ so that $X_i$ is like an $m+1$ sided die. Let ...