# 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/...
82 views

### 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, ...
1 vote
83 views

### 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 ...
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 $0\leq \theta \leq 1$ $L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2$ ...
61 views

### 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 ...
1 vote
33 views

### 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) \...
39 views

### 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....
1 vote
77 views

### 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 ...
1 vote
47 views

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 $x_{1}... 7 votes 2 answers 284 views ### What is the correlation between a random variable and its probability integral transform? Are there known bounds on the$\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$can maximize or ... 1 vote 1 answer 25 views ### Conditionalizing events on more than one event I am currently working on a question which seems to have an obvious answer, but it it seems just impossible for me to find a stringent proof of this relation (if it is true). Imagine the following ... 0 votes 0 answers 8 views ### Variance inequality for nested sets Let$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$are independent) then \begin{... 0 votes 1 answer 190 views ### Why is "Jensen's Inequality" Important in Probability? Why is Jensen's Inequality Important in Probability and Statistics? I was reading the Wikipedia page on "Convex Functions" (https://en.wikipedia.org/wiki/Convex_function), and came across ... 0 votes 0 answers 15 views ### For which probability distributions is this inequality true? I have derived the following inequality and now I want to check whether typical probability density functions exist, which fulfill this inequality. My suspicion is that it depends on the slope/... 0 votes 0 answers 76 views ### Would this extension of Khintchine's inequality be correct? This seems trivial, yet I have to make sure it is indeed correct. Referring to Roman Vershynin's High-Dimensional Probability book, the Khintchine's inequality (Exercise 2.6.5, page 27) is defined as ... 5 votes 2 answers 157 views ### How to calculate lower bound on$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|}{... 0 votes 0 answers 19 views ### 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? 2 votes 0 answers 72 views ### 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 \... 6 votes 3 answers 117 views ### 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 thatP(X_1>c)\le ... 2 votes 1 answer 37 views ### 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 ... 4 votes 1 answer 168 views ### Hoeffding type concentration result for the inverse of a sum of iid random variables Consider a collection ofn$i.i.d. Bernoulli random variables$\{ 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, $$\hat{\mu} = \... 1 vote 0 answers 58 views ### Holder's inequality in the case of L_1 and L_{\infty} norm I am referring to Wainwright's High-Dimensional Statistics book, where at some point it is deduced that \begin{equation} \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\... 9 votes 2 answers 3k views ### Proof that variance is always greater than or equal to zero It is common knowledge that:$$\begin{equation}\label{3} Var(X) \geq 0 \end{equation}$$for every random variable X. Despite this, I do not remember seeing a formal proof of this. Is there a proof ... 1 vote 0 answers 24 views ### When does this Deep Neural Network inequality become an equality? Let \text{DNN}_k(x) be some fully connected DNN with k hidden layers. Let (x,y) be some data points and \ell be a loss function. Then \forall \ k \in \{1,2,,..\} , we have that$$\min \ell(\... 2 votes 0 answers 42 views ### 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 ... 8 votes 1 answer 219 views ### 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 ... 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 ... 0 votes 0 answers 59 views ### 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,... 1 vote
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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 ...
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 $n$ ...
1 vote
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1 vote
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### 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\...
1 vote