1
vote
1answer
33 views

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 $x \in (0,1)$. ...
-2
votes
1answer
44 views

Markov/Chebyshev's inequality Problems

Let $X$ and $Y$ be two random variables for which $ E(X)=75 $, $ E(Y)=75 $, $\mathrm{var}(X)=10$, $\mathrm{var}(Y)=12$, $\mathrm{cov}(X,Y)=-3$ (i) Find and upper bound to $P(|X-Y| \ge ...
0
votes
1answer
61 views

Bias of expected binomial reciprocal

$X$ ~ Bionomial$(n, p)$. I want to evaluate the bias of $E\left(\frac{1}{X}\right)$,assuming $X>0$ (excluding the probability $(1-p)^n$), that is, the difference between $E\left(\frac{1}{X}\right)$ ...
0
votes
1answer
57 views

Inequality in statistics problem. Is my solution correct?

Probability of damaging an element in time $T$ equals $0.2$ How many such elements should there be so at least 50 won't be damaged after time T with probability 0.9, 0.95 and 0.99. So from what I ...
1
vote
0answers
42 views

Prove or disprove an inequality involving statistics

Do we have any result in statistics like this: $$|\overline x - \mu_e| \leq \sigma$$ Here $\overline x$ denotes the usual mean of some given discrete observations, $\mu_e$ their median and ...
0
votes
1answer
55 views

Which different probabilistic bounds/inequalities apply when we are given a lower bound on the sample size

Let m be the sample size and $X_i$ be a r.v. that we sample and define a new r.v. such that: $$M_m=\frac{1}{m}\sum^m_{i=1}{X_i}$$ My question is, what type of probabilistic inequalities require some ...
1
vote
0answers
106 views

How to use Chebyshev's inequality or the law of large numbers to a probability question

Let x be a random bit string that takes values $\{1,0\}^n$. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
0
votes
1answer
49 views

How to prove that $\frac{n}{\left(\sum\limits_{i=1}^n x_i\right)^2} \geq \frac{1}{\sum\limits_{i=1}^n x_i^2}$? [closed]

How to prove that $\frac{n}{\left(\sum\limits_{i=1}^n x_i\right)^2} \geq \frac{1}{\sum\limits_{i=1}^n x_i^2}$? I cannot seem to figure out why this holds. Any help would be much appreciated. Thanks.
0
votes
1answer
40 views

How to prove that the following inequality holds true?

To prove: $\frac{1}{2}(x_n-x_1)^2 < \sum\limits_{i=1}^n (x_i - \bar{x})^2 $. I simplified the LHS to: $\frac{1}{2} (x_n^2-2x_1+x_1^2)$ and the RHS to $\sum\limits_{i=1}^n x_i^2 ...
2
votes
0answers
135 views

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 ...
2
votes
1answer
69 views

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 ...
1
vote
1answer
126 views

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 ...
1
vote
1answer
78 views

Proof that $E[\exp(|A|)] \leq \exp(\delta) + \exp(\delta)P(|A|>\delta) + \int_\delta^\infty\exp(x)P(|A|>x)dx$

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) + ...
5
votes
1answer
188 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 $\bar{\lambda}=\frac{\sum_{i=1}^M\lambda_i}{M}$. ...
2
votes
2answers
217 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) &= ...
3
votes
0answers
97 views

Inequality of covariances between a bivariate normal vector and its indicator functions

Why holds for a standardized bivariate normal vector $Z:=(Z_1,Z_2)$ that \begin{equation} |\operatorname{cov}(Z_1,Z_2)|\geqslant |\operatorname{cov}(1\{Z_1\leq u\},1\{Z_2\leq u\})|? \end{equation} ...
4
votes
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: ...
1
vote
1answer
134 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}} ...
2
votes
1answer
367 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 ...
1
vote
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 $$ E\left|S\right|^q $$
2
votes
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 ...
1
vote
0answers
73 views

Inequality with Expectation and vectors

Let $x=(x_1,\ldots, x_n)\in R^n$. Let $r_i, i=1, \ldots, n$ be Rademacher i.i.d. random variables (i.e. $P(r_i=1)=P(r_i=-1)=1/2$). It is a well-known inequality that: $$ E\left(\left|\sum_{i=1}^n ...
2
votes
2answers
296 views

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 $x_{(n)} = \max_i x_i$ ...
1
vote
1answer
196 views

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?
1
vote
0answers
72 views

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 ...
0
votes
1answer
113 views

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 ...
2
votes
2answers
180 views

Inequality of Pearson correlation coefficient

Let $x_1,\ldots x_n,y_1,\ldots y_n$ be reals and $\bar{x},\bar{y}$ the aritmetic mean of numbers $x_1,\ldots x_n$ and $y_1,\ldots y_n$ respectively. How can I show that $$-1\leq \dfrac{\sum_{i=1}^n ...
4
votes
2answers
452 views

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 ...
2
votes
1answer
79 views

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 ...
0
votes
2answers
185 views

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: $$ P(\left| X-\mu \right| \ge k\sigma )\le \frac { 1 }{ { k }^{ 2 } } $$ ...
2
votes
2answers
110 views

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 ...
2
votes
1answer
220 views

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 ...
2
votes
1answer
69 views

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. ...
1
vote
1answer
70 views

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 ...
5
votes
2answers
4k views

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
5
votes
4answers
1k views

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 ...
4
votes
1answer
243 views

Interpretation of Markov's Inequality

Markov's inequality states that Pr(X >= t*E(X)) <= 1/t. This is great for asking "What is the probability that we get more than t times our expected value". ...
1
vote
3answers
96 views

can I get a bound on the probability of deviation, similar to Markov inequality?

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 ...