# 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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Let $A$ be a random variable and $P$ be a probability measure. For some real $\delta>0$, is there a simple proof that $E[\exp(|A|)] \leq \exp(\delta) + \exp(\delta)P(|A|>\delta) + ... 1answer 187 views ### Jensen's inequality I am using Jensen's inequality and conditional expectation to prove the following inequality: Let$\lambda_i$be real for$i\in \{1,2,...,M\}$and$\bar{\lambda}=\frac{\sum_{i=1}^M\lambda_i}{M}$. ... 2answers 215 views ### Distance between the product of marginal distributions and the joint distribution Given a joint distribution$P(A,B,C), we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= ... 0answers 97 views ### Inequality of covariances between a bivariate normal vector and its indicator functions Why holds for a standardized bivariate normal vectorZ:=(Z_1,Z_2)$that $$|\operatorname{cov}(Z_1,Z_2)|\geqslant |\operatorname{cov}(1\{Z_1\leq u\},1\{Z_2\leq u\})|?$$ ... 2answers 51 views ### Integral inequation In my statistics book Chebyshev's inequality is proven. In several steps this inequality is used: $$\int_a^{+\infty} \phi(x) f_X(x)dx \quad \geq \quad \phi(a) \int_a^{+\infty} f_X(x)dx$$ and also: ... 1answer 129 views ### Chernoff bound proof using Markov Does anyone familiar with the following format of Chernoff bound: $$Pr\left(\frac{1}{n}\sum\limits_{i=1}^n X_i \gt T\right ) \le \inf_{\gamma \gt 0}{\left ( \frac{E[e^{\gamma X_i}]}{e^{\gamma T}} ... 1answer 359 views ### Proof of Frechet-Hoeffding Copula bounds How is the lower Frechet-Hoeffding copula bound proved? In the bivariate case, it follows from C(u_1,u_2)-C(u_1,v_2)-C(v_1,u_2)+C(v_1,v_2)\geq0 by setting (v_1,v_2)=(1,1). I'm struggling to ... 1answer 98 views ### estimation of a moment for the sum with Bernoulli random variables Let x\in R_+^n and let b_i, i=1, \ldots, n be (0,1) Bernoulli random variables with P(b_i=1)=p. Denote S=\sum_{i=1}^n x_ib_i. For q\geq 2 estimate from above$$ E\left|S\right|^q $$1answer 50 views ### Prove \sum_{i=1}^n$$w_i^2\geq\frac{1}{n}$ given $\sum_{i=1}^n w_i=1$

I was looking at my stats textbook and they claim that the sample variance of a weighted distribution involving i.i.d. $x_i$s will be smallest when each of the weights is equal. I follow this argument ...
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### Cauchy-Schwarz matrix 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 $X$ and $Y$ are random vectors, is there a way to bound ...
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### Proving or refuting an inequality regarding the variance

I'm trying to prove, or find a counterexample, for the following problem: Let $Y = \{y_i\}_{i=1}^n$ be a set of data, where $y_i \ge 1$ for $i \in \{1,\ldots,n\}$, and let $\alpha$ be a natural ...
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### Result of Chebyshev's Inequality if just more than instead of more than or equals?

This is a question that I happen to think of when looking at the Chebyshev's Inequality. In the inequality, it has this: $$P(\left| X-\mu \right| \ge k\sigma )\le \frac { 1 }{ { k }^{ 2 } }$$ ...
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### How can I prove this inequation $\Pr\{X+Y<t\} \le \Pr\{X<t\} \Pr\{Y<t\}$

Could you please help me to prove the inequality probability as follows: $\Pr\{X+Y<t\} \le \Pr\{X<t\} \Pr\{Y<t\}$ where $X$ and $Y$ are non-negative independent random variables with common ...
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### Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$, $x_{1}\ge\cdots\ge x_{10}>0$ and ...
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### Incomplete “round trip” of taking a minimum, then a maximum, from a positively skewed distribution

Let's say you have a distribution that is either symmetric or positively skewed (and defined over 0-1). Call it F. Then, you find the distribution of the minimum of n>1 draws from F. Call it Fmin. ...
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### Can one prove $\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min(1,\frac{c}{t})$?

Let $c>1/2$ be an arbitrary big fixed constant. Can one prove that for all $t\geq 1$: $$\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min\left(1,\frac{c}{t}\right)$$ for some small constant ...
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### Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$\mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>.$$
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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 ...