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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Cauchy Schwarz inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
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An inequality involving expectation

Let $f,g$ be two pdfs, and suppose $X$ is a random variable that has pdf $f$. Is it necessarily true that $E[f(X)] \ge E[g(X)]$? Although I doubt this will help, but I got this problem from studying ...
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How prove this distributions inequality $cov(\theta_{i},\theta_{j})\ge 0$?

Question: let random variable $\theta$ has dendity $f_{\phi}(\phi)$,and the random vector $\theta=(\theta_{1},\theta_{2},\cdots,\theta_{n})$,such $\theta_{i}|\phi$ are all independent from each ...
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Upper bound for the absolute value of an inner product

I am trying to prove the inequality $$\left|\sum\limits_{i=1}^n a_{i}x_{i} \right| \leq \frac{1}{2}(x_{(n)} - x_{(1)}) \sum\limits_{i=1}^n \left| a_{i} \right| \>,$$ where $x_{(n)} = \max_i x_i$ ...
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the upper and lower bound of the max of two iid variables

If $x$ and $y$ are independent standardized random variables with zero mean and unit variance, what are the upper and lower bounds of $$E[\max(x,y)]$$ Can I get a useful reference to this?
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Reference for Khinchine inequality

I am looking for the proof of Khinchine inequality (see http://en.wikipedia.org/wiki/Khintchine_inequality for example), using martingales and the Azuma inequality. Can you please help me to find a ...
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How do I find these bounds using Chebychev's inequality and the Central Limit Theorem?

Let $X$ have gamma distribution with parameters $\alpha=7$ and $\lambda=1$. Investigate the value of $F_X(10)$ using these methods: Find a lower bound using Chebychev's inequality. Approximate the ...
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Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
I have two random variables $X$ and $Y$, both receiving values between 0 and 1. I know that $E[X - Y] \ge 0$. Can I get any inequality of the form: $P(X - Y \ge \delta) \le F(\delta,X,Y)$ where ...