# Questions tagged [probability-inequalities]

Probability Inequalities are useful for bounding quantities that might otherwise be hard to compute. A related concept is a concentration inequality, which specifically provides bounds on how far a random variable deviates from some value.

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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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### Bounds on distance between two independently variables drawn from the same distribution

Suppose $X_1$ and $X_2$ are iid from an arbitrary distribution with variance $\sigma^2$. How can we derive an upper bound for: $$P(|X_1-X_2|\ge\delta)$$ One simple idea is Chebyshev's Inequality, ...
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### Convergence in probability and Chebyshev inequality

Given problem: The elegant solution is to use Markov inequality for $X^2_n$. But my solution was via Chebyshev inequality, smth like that: $P(|X_n - 1/n| \ge k) \le \frac{\sigma^2}{k^2}$, now lets ...
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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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### 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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### Applicability of Hoeffding's Inequality

I am working through Larry Wasserman's All of Statistics. I am trying to understand why Hoeffding's Inequality was valid for the following problem: Suppose we test a prediction method on a set of $n$ ...
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### Covariance of X and Y conditional on X+Y>Z? [closed]

Suppose that $X$, $Y$, and $Z$ are three independent random variables. Is there a way to compute the following conditional covariance? $Cov(X, Y | X + Y \geq Z)$
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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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### I've taken an alternative approach to a convergence in probability problem. Is there any mistake and/or is my conclusion correct?

I am currently studying convergence on my own, which means that I don't have many alternatives for discussing problems in order to improve my understanding. This post was an alternative to get around ...
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### Can we bound $\frac{Cov(X,XY)}{Var(X)}$?

The question is can we bound $\beta = \frac{Cov(X,XY)}{Var(X)}$ with the help of the following assumptions : Y is a positive bounded random variable, let's assume $Y \in [0,1]$. X has an expectation ...
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### Symmetrization in Proof of Hoeffding's Lemma

This alternative proof of a slightly weaker version of Hoeffding's Lemma features in Stanford's CS229 course notes. What's notable about this proof is its use of symmetrization. However, I find this ...
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### Why $Pr[X-\mu \geq t]= Pr[e^{\lambda(X-\mu)} \geq e^{\lambda t}]$ for all $\lambda> 0$

I hope everyone is having a nice day. I don't know why this inequality holds. $$Pr[X-\mu \geq t]= Pr[e^{\lambda(X-\mu)} \geq e^{\lambda t}]$$ For $\lambda >0$. I guess it has something to do ...
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### How to select a set of MCMC samples with some probability?

I have a set of samples gathered using MCMC of random variable $X$. Let's call this set $X_s$. How to select a subset of samples $S$such that $$Probability(X_s(S) < constant) > 0.01$$ I want ...
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### How to derive Bonferroni's Inequality using Boole's Inequality?

I'm trying to derive Bonferroni's inequality using : $$P(\cup^{\infty}_{i=1} A_i) \leq \Sigma^{\infty}_{i=1} P(A_i)$$ for any sets A_1, A_2, ... (Boole's Inequality) The result I want is (Bonferroni'...
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### Connection between subgaussian/subexponential and exponential family

I am wondering if there is any relationship between subgaussian/subexponential with (one parameter) exponential family. In particular, is there any sub-family density that belongs to both ...
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### Bound on sample size- Hoeffdings inequality

Studying for my upcoming statistics exam I tried to solve the following: In some population, each individual likes exactly one out of 30 possible music genres. In some survey, n people are drawn ...