# 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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### 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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### Question about an implication relationship

Let $||\cdot||$ be the Euclidean norm. Suppose $X_1,X_2$ are two independent and identically distributed random variables, and $a_N(X_1,X_2)$ is a vector valued function that depends on factor $N$ and ...
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### Assessment of bias by imputation

Lets say i have a large survey dataset which i want to use as a source for income reporting of the population, e.g. parameters of the distribution, poverty and inequality. Due to item nonresponse on ...
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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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### 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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### 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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### Creating a transition matrix based on a Markov chain in R

I have four distributions that represent incomes in R. I categorise them by what income group they fall under such as under half the mean, between half the mean and 3/4th of the mean and so on until ...
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### Proving two different expressions of non-centrality parameters are equivalent

I am stuck in proving $$\sum_{i=1}^{K}\xi_i(\mu_i - \bar{\mu})^2 = \sum_{i,j}\xi_i\xi_j(\mu_i - \mu_j)^2,$$ where $\bar{\mu} = \sum_{i=1}^{K}\xi_i\mu_i$ and $\sum_{i=1}^{K}\xi_i = 1$. I am not sure ...
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### Variance inequality

Why does the following hold? For a random variable X with finite necessary moments, $E(|X|) \leq \sqrt{Var(X)}+|E(X)|$
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### Does an inequality hold as an expectation over a probability distribution?

Suppose I have to functions $f(x)$ and $g(x)$ such that $$f(x) \leq g(x) \quad \forall x.$$ For a distribution $\pi(x)$ on $x$, is it necessarily true that $$E_\pi[f(x)] \leq E_\pi[g(x)]?$$ My ...
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### A different proof for KL divergence non-negativity

KL divergence's non-negativity can be proved in many ways. One could use the inequality $\log x \leq x - 1$ as a main step in the proof, another one could leverage the property of concave of the ...
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### Cauchy Schwarz inequality proof using discriminant

I know the proof but I'm unclear on one thing. Cauchy-Schwarz inequality: Given X,Y are random variables, the following holds: $$(E[XY])^2 \le E[X^2]E[Y^2]$$ Proof Let $$u(t) = E[(tX - Y)^2]$$ ...
### Prove that $E(X\ln X)\le E X E\ln X$ [closed]
I want to prove it using Jensen inequality, so I need to prove that $g(x)=x\ln x$ is a convex function, which means $$g\left(\frac{a+b}{2}\right)\le \frac{1}{2}\left(g(a)+g(b)\right).$$ How can I ...