Questions tagged [inequality]

Use this tag if you question involves the use of an inequality. The inequality may have probabilistic origins or be a purely mathematical inequality. Do not use for measures of inequality, for instance income inequality. For that use the [diversity] tag.

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Probability of higher hourly earnings by gender from UK data [duplicate]

I'm trying to find the probability that in the UK, in a male, female pair picked at random, the female will have the higher annual income. I have following data from https://www.ons.gov.uk/...
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What does it mean that the decomposition is based on the linear systematic component? And how can I interpret my result?

I'm using the oaxaca package to implement a Blinder-Oaxaca decomposition on a logistic model with binary outcome. The vignette says that: Note that, if a non-linear function such as glm() is chosen, ...
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2 answers
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Sufficient condition for $ \sigma_{X}^{2} \leq \sigma_{Y}^{2}$

Suppose $X$ and $Y$ are random variables whose expected values are $\mu_X$ and $\mu_Y$, and variances are $\sigma_{X}^{2}$ and $\sigma_{Y}^{2}$, respectively. Also, we suppose $F_x$ and $F_Y$ are the ...
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5 votes
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298 views

Taylor expansion in Hoeffding's Lemma proof

Hoeffding's Lemma proof uses Taylor expansion with this statement: From Taylor's theorem, for some $ 0\leq \theta \leq 1$ $ L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2 $ ...
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What is the difference between MVB UMVUE and MVUE.?

Cramer Rao inequality gives MVB and if MVB exist it is MLE. Rao Blackwell gives UMVUE, but isn’t when we have MVB estimator for unbiased it is UMVUE? Then what is MVUE? MVB minimum variance bound ...
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For $Y \geq 0$, prove that $Pr(Y \geq k) \leq E(Y)/k$

Let $Y$ be a non-negative random variable, $k$ be any positive constant, show that $Pr(Y \geq k) \leq E(Y)/k$. My attempt (using integration by parts): \begin{align} \int_0^k y \,dF(y) &\leq E(Y) \...
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Sum of values with different probabilities

Suppose I have the following linear expression: $S = x_1 + x_2+ \dots + x_n$, in which each $x_i$ can only assume the following values: -2, -1, 0, 1, 2 whose probabilities are 0.1, 0.2, 0.2, 0.25, 0....
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Probability Conditioned on Inequality

Assume that $A \sim \mathcal{N}(0, 1)$, $B \sim \mathcal{N}(0, 1)$. I am trying to calculate $P(A \,|\, A < B)$. For the sake of this problem, we can assume that $A \perp B$, but (for obvious ...
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Khintchine inequality for the linear combination of sparse Bernoulli random variables

Let $\{\epsilon_{n}\}_{n=1}^{N}$ be i.i.d. random variables with $P(\epsilon_{n} = \pm 1) = 1/2$ for $n=1,2, \ldots, N$ i.e. a sequence of Rademacher distribution. Let $0<p<\infty$ and let $x_{1}...
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What is the correlation between a random variable and its probability integral transform?

Are there known bounds on the $\operatorname{cor}(X,F(X))$? $X$ is a random variable with CDF $F(X)$. Let $X$ have a fixed variance, for example $\operatorname{var}(X)=1$. What $X$ can maximize or ...
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Conditionalizing events on more than one event

I am currently working on a question which seems to have an obvious answer, but it it seems just impossible for me to find a stringent proof of this relation (if it is true). Imagine the following ...
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Variance inequality for nested sets

Let $X, Y,$ and $Z$ are three random variables/vectors, and let $f(., ., .)$ is a real-valued, deterministic function. If $Z$ is independent of $\{X, Y\}$ (e.g., $X, Y, Z$ are independent) then \begin{...
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Why is "Jensen's Inequality" Important in Probability?

Why is Jensen's Inequality Important in Probability and Statistics? I was reading the Wikipedia page on "Convex Functions" (https://en.wikipedia.org/wiki/Convex_function), and came across ...
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For which probability distributions is this inequality true?

I have derived the following inequality and now I want to check whether typical probability density functions exist, which fulfill this inequality. My suspicion is that it depends on the slope/...
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Would this extension of Khintchine's inequality be correct?

This seems trivial, yet I have to make sure it is indeed correct. Referring to Roman Vershynin's High-Dimensional Probability book, the Khintchine's inequality (Exercise 2.6.5, page 27) is defined as ...
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How to calculate lower bound on $P \left[|Y| > \frac{|\lambda|}{2} \right]$?

Let $Y$ be a random variable such that $E[Y] = \lambda$, $\lambda \in \mathbb{R}$ and $E[Y^2]<\infty$. The problem is to find a lower bound on the probability $$ P \left[|Y| > \frac{|\lambda|}{...
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Statistical Test for Some Inequality Condition

Suppose that we have three random variables, named $V_1, V_2$ and $V_3$. Here, I want to test the following inequality: $V_1 \geq V_2, V_3$ Is there any reference dealing with this topic?
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Problem understanding the intuition behind Slepian's inequality

Slepian's inequality is defined as follows: Let $X\in\mathbb{R}^n$ and $Y\in\mathbb{R}^n$ be centered Gaussian random vectors such that \begin{align} \mathbb{E}X_iX_j&\geq \mathbb{E}Y_iY_j,\quad \...
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6 votes
3 answers
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Is $P(|X_1|>k)\le P(|X_2|> k)$ when $X_i\sim N(\mu_i,\sigma^2)$ and $|\mu_2| \ge |\mu_1|$?

Suppose $X_1\sim N(\mu_1,\sigma^2)$ and $X_2\sim N(\mu_2,\sigma^2)$ where $\mu_2\ge \mu_1$. Since $\mu_2\ge \mu_1$, based on a characterization of stochastic ordering, we can say that $$P(X_1>c)\le ...
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Find the minimum percentage of values within two bounds subject to moment constraints

I am facing the following problem: Variable Z has a mean of 15 and a standard deviation of 2. What is the minimum percentage of Z values that lie between 8 and 17? I have tried the following: Here ...
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Hoeffding type concentration result for the inverse of a sum of iid random variables

Consider a collection of $n$ i.i.d. Bernoulli random variables $\{ X_i \}_{i=1}^{n}$ with $\mathbb{E}[X_i] = \mu$. Then, if $\hat{\mu}$ is the mean of the $n$ random variables, i.e. if, $$\hat{\mu} = \...
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Holder's inequality in the case of $L_1$ and $L_{\infty}$ norm

I am referring to Wainwright's High-Dimensional Statistics book, where at some point it is deduced that \begin{equation} \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\...
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9 votes
2 answers
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Proof that variance is always greater than or equal to zero

It is common knowledge that: $$\begin{equation}\label{3} Var(X) \geq 0 \end{equation}$$ for every random variable $X$. Despite this, I do not remember seeing a formal proof of this. Is there a proof ...
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1 vote
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When does this Deep Neural Network inequality become an equality?

Let $\text{DNN}_k(x)$ be some fully connected DNN with $k$ hidden layers. Let $(x,y)$ be some data points and $\ell$ be a loss function. Then $\forall \ k \in \{1,2,,..\} $, we have that $$\min \ell(\...
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2 votes
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prove the difference between mean and median is less than the variance [duplicate]

Suppose $X$ is a random variable with finite variance. Let $m$ denote the median of $X$ and $\mu$ the mean of $X$, i.e. $\mu=\mathbb{E}(X)$. Show $$(m-\mu)^2\leq\text{var}(X)$$ Intuitively this is ...
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8 votes
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Prove that Kurtosis is at least one more than the square of the skewness

Wikipedia claims it, and on reading the paper that it linked I found that the proof that was written there was quite difficult. Is there a simple proof possible for this identity? The proof given in ...
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4 votes
1 answer
244 views

Upper bound for absolute third central moment

Suppose $X\in \mathbb{R}$ is a random variable with expected value $\mathbb{E}X = \mu$. I ran across a proof which uses the inequality $$ \mathbb{E}[|X - \mu|^3] \leq 2^3 \mathbb{E}|X|^3. $$ Can ...
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cauchy schwarz inequality on sum of squares

Can I and how I can use the Cauchy Schwarz inequality on the amplitude of a imaginary sum of squares? $$Z = X+iY$$ and $$|Z| = \sqrt{X^2 +Y^2}$$ to show that $$|E[Z] | \leq \mathbb{E}[|Z|]$$ where $X,...
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1 vote
0 answers
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How can I weight ordinal observations and reduce them to one statistic? [closed]

It will be a complicated question and I try to briefly explain. I am studying on educational inequalities. The survey I use for analysis incudes ordinal variables and the education degree which ...
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3 votes
1 answer
28 views

Implications of zero limiting variance

Assume that I have a sequence of random variables $X_1, X_2, \dots$ with means $\mu_1, \mu_2, \dots$ such that $\lim_{n \to \infty} \operatorname{Var}(X_n) = 0$. Can I claim that for large enough $n$ ...
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1 vote
0 answers
305 views

Proof of the multivariate Cramer-Rao inequality

I search a detailed proof of the multivariate Cramer-Rao inequality in the general case where the estimator is not necessarily unbiased. Let $T(X)$ be an estimator of the parameter $\theta\in\mathbb{R}...
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2 votes
0 answers
24 views

How to test that a sequence of variances rank ascendingly?

I am investigating forecast optimality. Diebold (2017, p. 334, list item d) indicates that one of the desirable properties of a good forecast is Optimal forecasts have $h$-step-ahead errors with ...
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1 vote
1 answer
353 views

Chebyshev's inequality for Pareto distribution (3 sigma rule)

According to the Chebyshev's inequality, if we take any distribution, we get >88.8889% of data in +-3 sigma interval. For a normal distribution it is 99.97%. How to calculate the interval for a ...
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3 votes
2 answers
90 views

Boundary of $E\left[\frac{\prod_{i=1}^n x_i}{\prod_{i=1}^n x_i+\prod_{i=n+1}^m x_i}\right]$

Suppose $X_i$ are i.i.d. In addition, $X_i>0$ and $E[X_i]>1$. Suppose $E[X_i]$ is known, could we find upper bound or lower bound for the following expectation: $$ E\left[\frac{\prod_{i=1}^n x_i}...
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3 votes
1 answer
378 views

When is Jensen's Inequality strict?

For a homework problem, I have to prove that for a random sample $X_1, \ldots, X_n$, drawn from a population with finite variance $\sigma^2$, with sample mean $\bar{x}$ and sample variance $s^2$, that ...
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0 votes
1 answer
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Question about property of 2-increasing, grounded function with margins

The question is about what may very well be an obvious detail in the proof of a lemma from the book An Introduction to Copulas by Roger Nelsen. I will state all the relevant results and definitions ...
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2 votes
1 answer
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Distributive property of probabilistic inequalities involving random variables on both sides

Can I break down $P(h \geq (A + B)$, given all $ A,B,h$ are all random variables. Will the following rule works? $$P[h \geq (A + B)] = P(h\geq A) + P(h\geq B)$$ Actually, in one of my mathematical ...
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Bounding the norm of the difference between two related probability densities

Suppose we have a continuous random variable $X$ and two continuous functions $f$ and $g$ such that $f(X)$ and $g(X)$ are continuous random variables. Let $p_A$ be the probability density function of ...
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1 vote
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Generalization of the Payley-Zigmund inequality

The Payley-Zigmund inequality states that for a positive random variable $Z$ the following holds \begin{equation} \operatorname{P}( Z > \theta\operatorname{E}[Z] ) \ge (1-\theta)^2 \frac{\...
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1 vote
0 answers
37 views

Are this simple claim and its proof correct?

Suppose random sequence $\{X_{i}(N)\}_{i=1}^{N}$ is a row-wise i.i.d. triangular array, where $N$ is sample size. This means for any given $N$, $X_{i}(N),\dots,X_{N}(N)$ are i.i.d. following ...
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3 votes
1 answer
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Use Chebyshev's inequality to find a lower bound of a Chi-Square Distribution

I'm trying to solve the following exercise but I'm not sure if what I'm doing is right. "Let $X$ be an r.v. distributed as $\chi_{40}^{2}$. Use Tchebichev’s inequality in order to find a lower ...
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1 vote
1 answer
243 views

Correlation Coefficient Squared is Less Than or Equal to One

Problem Statement: Let $Y_1$ and $Y_2$ be jointly distributed random variables with finite variances. Let $\rho$ denote the correlation coefficient of $Y_1$ and $Y_2.$ Using the inequality $$[E(Y_1Y_2)...
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How important is triangle inequality for statistical estimators?

(Pearson's) correlation is a measure of co-dependence that does not fulfill certain axioms such non-negativity and triangle inequality. In layman's terms, how would you describe what triangle ...
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2 votes
1 answer
78 views

Upper Bound and Lower Bound on Means when Distributions are bounded?

Suppose we have two different probability distributions $p, q$ defined on input $x \in [0,1]$. We know that for any value of $x$ in the domain, we have $\exp^{-a} \leq \frac{p(x)}{q(x)} \leq \exp^{a} $...
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0 votes
1 answer
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Laplace Inequality

I am trying to prove that if $r_i \sim Lap(0,1/\varepsilon)$ where $\varepsilon >0$ then: $$Pr[r_i \geq 1+r^*] \geq e^{-\varepsilon}Pr[r_i \geq r^{*}]$$. I know that for $r*>0$ it satisfies ...
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0 votes
1 answer
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$var(y)=b^2var(x)+var(e)$

Suposse $y=xb+e$ where $y$ and $x$ are random variable. $e$ is the error of the regression. Since x and e are independent then: $var(y)=b^2var(x)+var(e)$ How can I proof the following double ...
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0 answers
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$E(xy)<\infty$ proof

I am reviewing the best linear projection properties proof in Hansen's book on econometrics. Specifically, the proof according to which $E(xy)<\infty$. For this, it is assumed that $E(y^2)<\...
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1 vote
0 answers
19 views

inequality involving mean, median and variance [duplicate]

I'm looking to show $|{\rm med}(x)-\bar{x}|\le{\rm sd}(x)$. I did a bunch of simulations and the statement seems right to me. $$ {\rm Var}(x)=\frac{1}{n}\sum\left(x_i-\bar{x}\right)^2=\frac{1}{n}\sum\...
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1 vote
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254 views

REINFORCE algorithm, help for the proof of the variance reduction by subtracting a baseline

I'm trying to find a proof or an approximate argument justifying that, in the REINFORCE algorithm, subtracting a baseline to the episode reward reduces the variance. I believe this proof can be done ...
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1 vote
0 answers
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How to include information from observations with mathematical inequalities in Ordinary Least Squares regression?

So, I was using Ordinary Least Squares (OLS) linear regression to build a model describing pond water level fluctuations in function of precipitation and potential evapotranspiration (PET) data. The ...
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