Tagged Questions

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How to show that $\Phi(1-x)^{-1} =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
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Finding conditions for joint probability density larger than the product of marginals

I was wondering if you could help me out. I have a joint probability distribution with density $f(x,y)$ and marginals $g(x)$ and $h(y)$ defined over the real line. Now, I would like to find a class of ...
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Inequality with moments

Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here). Is $m(f^3) \le m(f^2) m(f)$?
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Motivation behind steps in proof of Hoeffding Inequality

The lemma that is proved for proving Hoeffding's inequality is: If $a\leq X\leq b$ and $E[X]=0$, $E[e^{tX}] \leq e^{\frac{t^2(b-a)^2}{8}}$ Here's a link to the proof: ...
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Bell's inequality

Let $\xi, \eta, \zeta$ be random variables such that $|\xi|, |\eta|, |\zeta| \le 1$. I need to prove such inequality: $|\mathbb{E}(\zeta \xi)-\mathbb{E}(\zeta \eta)| \le 1 - \mathbb{E}(\xi \eta)$ ...
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Is it true $({\Bbb E|X|^p})^{\frac{1}{p}} \leq ({\Bbb E|X|^q})^{\frac{1}{q}}$, if $p \leq q$?

I ran into this post which shows $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$, $p \leq q$. So I guess it's true $({\Bbb E|X|^p})^{\frac{1}{p}} \leq ({\Bbb E|X|^q})^{\frac{1}{q}}$, if $p \leq q$, ...
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Why $X \geq 0$ and $\Bbb E{X} < + \infty$ implies that $\lim_{y \to 0^{+}}y \Bbb{E}{(\frac{1}{X} | X > y)} = 0?$

Why $X \geq 0$ and $\Bbb E{X} < + \infty$ implies that $\lim_{y \to 0^{+}}y \Bbb{E}{(\frac{1}{X} | X > y)} = 0?$ I'm thinking about replacing $\Bbb{E}{(\frac{1}{X} | X > y)}$ with ...
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How prove $E|x|^p<+\infty,E|y|^p<+\infty$ ,if $E|x+y|^p<+\infty$
let two random variables X and Y are independent of each other,for some $p>0$,we have $$E|x+y|^p<+\infty$$ show that $$E|x|^p<+\infty,E|y|^p<+\infty$$ my try: I know Minkowski inequality ...
If I know that $EX^4=1$ and $EX\leq0$, how do I get an upper bound for $EX^3$? I know that by Jensen's inequality, $EX^3\leq1$, but I need the upper bound less than 1.