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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33 views

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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20 views

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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22 views

Expectation of (sum subtract the expectation of sum)

Let's say we have random variables $\mathbf{X}$, and we have $P(\mathbf{X}\in [a, b])=1$, we have $\mathbf{S}_n = \mathbf{X}_1 + \mathbf{X}_2, +\dots + \mathbf{X}_n$. If $\mathbf{X}_1, \mathbf{X}_2, ...
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Is there a different way to bound probability, with no distributional assumptions, other than Markov's inequality?

I'm having trouble answering this probability interview question from Interview Query: Let's say there is man who is 5.10 ft tall who doesn't know how to swim. Let's say he wants to swim in a lake ...
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121 views

Tail bounds for sample means of i.i.d random variables where the moment generating function exists

I want to figure out the proof of Lemma 4 in the following paper. The lemma states that Let $Y_1, Y_2,\ldots Y_m$ be i.i.d random variables such that $\mathbb{E}\left[e^{zY}\right] < \infty$ ...
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27 views

How to using the Markov Inequality to find the upper bound for $\mathbb{P}(X > 2)$ given I only have information about $X^4$?

Let $X$ be a nonnegative random variable that satisfies $\mathbb{E}[X^{4}]=4$ . How should I calculate an estimate for the $\mathbb{P}(X \geq 2)$ using the Markov Inequality? I tried to find a ...
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Formula of the Chebyshev's inequality for an asymmetric interval

The formula for Chebyshev's inequality for the asymmetric two-sided case is: $$ \mathrm{Pr}( l < X < h ) \ge \frac{ 4 [ ( \mu - l )( h - \mu ) - \sigma^2 ] }{ ( h - l )^2 } . $$ What I don't ...
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Showing the weak law of large number of non-IID sequence of random variables

Common proofs on the law of large numbers usually assume a sequence of IID random variables. If $X_1\dots X_n$ has a common expected value $\mu$, finite but not necessarily common variance (hence not ...
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Using Hoeffding's inequality on sum of uniform variables

I have the following problem: $X_1,...,X_n$ are i.i.d. $\sim U(-3,5)$ continuous uniform variables in the support between -3 and 5. $S := X_1 + ... + X_n$. I need to use Hoeffding's inequality to ...
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43 views

Basic probability inequality between 4 events

I have been asked to prove the following but am unsure if it is true: $$\mathbb{P}(A,B,C)<\mathbb{P}(D) \implies \mathbb{P}(A) + \mathbb{P}(B) + \mathbb{P}(C) - 2 < \mathbb{P}(D)$$ I don't ...
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27 views

Are the following terminologies error/risk/marmgin/regret bounds related?

I recently come across papers with titles resembling "Error/Risk/Margin/Regret Bounds" and I can't help but wondering if there is any fundamental (mathematical) difference between these terminologies? ...
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179 views

Jensen inequality and bias of sample standard deviation

I am currently studying Introduction to Probability, second edition, by Blitzstein and Hwang. In studying the Jensen inequality, the following example is presented: Example 10.1.6 (Bias of sample ...
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How close a sample is to the Normal distribution ( Berry-Esseen Theorem)

My question is how can I use the Berry-Esseen Theorem to know how close to the Gaussian distribution is $L$, where $$L=nLn(2)+Ln(r_1)+Ln(r_2)...Ln(r_n).$$ $r_i \geq 0$ is a i.i.d. random variable ...
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What does the general case of bounded random variables mean in terms of Hoeffding's Inequality?

The following equation is Hoeffding's Inequality from Wikipedia for the general case of bounded random variables. I have just come to understand Hoeffding's Inequality for the special case of ...
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Need mathematical steps for Hoeffding's Inequality applied to Bernoulli Distribution

I am trying to understand Hoeffiding's Inequality in Machine Learning and I am referring to WikiPedia for it. Hoeffding's Inequality is defined as follows: $ P(|\hat{\theta} - \theta)| \ge \epsilon) \...
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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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38 views

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 ...
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Hoeffding's inequality vs DKW inequality

What is the difference between Hoeffding's inequality: $$\mathrm{P}(|\hat{F}_n(x)-F(x)|\geq \epsilon) \leq 2e^{-2n\epsilon^2}$$ and the Dvoretzky-Kiefer-Wolfowith (DKW) inequality: $$\mathrm{P}(\...
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Prove that $E[e^{2(m−1)X^2}]\le m$

I'm reading "Understanding Machine Learning: From theory to algorithms". The problem is as follows, which is Exercise 31.1 of the book on page 416. Let $X$ be a random variable that satisfies $P[X \...
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Show that if $Y$ has $E(Y)=μ,Var(Y)=σ^2$, and $P(|Y-μ|<σ)=0$, then $Y$ has the same distribution as $X$ described below for $k=1$

I'm having a hard time proving the second and third case of $X$: Show that if $Y$ has $E(Y)=μ,Var(Y)=σ^2$, and $P(|Y-μ|<σ)=0$, then $Y$ has the same distribution as $X$ described below for $k=1$. $...
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How can I show that $P\{|(X-\mu_X)+(Y-\mu_Y)| \ge k\sigma\} \le (2(1+\rho))/k^2$?

Let $\sigma^2$ be the common variance of the random variables $X$ and $Y$, with their correlation coefficient being $\rho$. Show that $\forall k>0$, $P\{|(X-\mu_X)+(Y-\mu_Y)| \ge k\sigma\} \le (2(...
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18 views

Markov inequality and Boundness in probability

Let $\{X_n\}$ and $\{a_n\}$ be sequences of random variables and real numbers, respectively. Say that $X_n=O_P(a_n)$ iff $\forall\epsilon>0$, $\exists N,M>0$ such that for all $n>N$, we ...
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23 views

Does this expectation inequality holds?

Let $X\in L_p(P), p>1$. Is the following result true? $$E[\lvert X\rvert I(\lvert X\rvert>C)]\leq C^{1-p}E\lvert X\rvert^p.$$ where $C>0$. It can be found in the proof of Corollary A.1 (...
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Exponential Inequality For Probability of Being Close to Maximum

Given $n$ independent identically distributed random variables $X_1, X_2, \ldots, X_n$ that have $|X_i| < \lambda$ for all $i$. Let $\max(X)$ be the maximum of these $n$ variables. Is there a ...
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41 views

Can Markov inequality be used to define bounds in a meaningful way?

Suppose $X\sim \text{Binomial}(100,0.5)$. Recall Markov's inequality. $$\Bbb{P}(X\geq 5) \leq \frac{\Bbb{E}[X]}{5}=\frac{100*0.5}{5} = 10$$ Why is this inequality valuable? Since I'm working with a ...
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What is the probability that maximum likelihood algorithm is right in Bernoulli trials?

Let $\varepsilon\in(0,1)$ and $p:=\frac{1+\varepsilon}{2}$. Suppose that we have a sequence of independent Bernoulli random variables of parameter $p$, say $(X_k)_{k\in\mathbb{N}}$ defined on a ...
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77 views

Sum bounded in probability

Suppose that $\sum_{n=1}^N|c_n| = O(1)$ and that $X_n = O_p(1)$ in the sense that for every $\epsilon>0,\exists M<\infty$ such that $$\sup_nP(|X_n|>M)<\epsilon$$ Can we claim that $\sum_{...
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32 views

Usefulness and validity of Alternative definitions of “quantile”

According to textbook, the $p\,$th quantile of a random variable $X$ is any real value $x$ satisfying $P(X \geq x)\geq 1-p$ and $P(X \leq x) \geq p$. Why isn't the alternative definition, a $p\,$th ...
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A consistent estimator with infinite expectation?

Typical (or common) approaches to prove an estimator is consistent require finite mean and variance. The proofs usually follow from concentration bounds, e.g. Markov, Chebyshev, etc. I'm wondering ...
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Probability of sparse spectrum

Consider a vector $v$ such that $v \sim \mathrm{Unif}(\mathbb{S}^{d-1})$, the uniform distribution on the unit sphere in $d$ dimensions. Question: is there an upper bound on the probability that $v$ ...
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40 views

Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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58 views

Unusual Markov inequality for normal distribution

I'm trying to answer the following question from Larry Wassermans book on statistical inference. My question is how did they arrive at the Markov bound, it does not seem like the normal form of the ...
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21 views

probability density of a random vector greater than some value? [duplicate]

In single dimension, the probability that a random variable $X$ is greater than some value $x$ is easily related to the cumulative distribution(c.d.f.) as $Pr(X > x) = 1 - F(x)$ if only $Pr[X \leq ...
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Estabilishing an upper bound for the probability of an impossible event, by sampling [duplicate]

Lets suppose there is an event that gives a random outcome each time it happens. The set of possible events is finite, but their probabilities differ, sometimes by orders of magnitude. (imagine a ...
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82 views

How can I prove $ P(X> 0) \geq \frac{(E[X])^2}{E[X^2]}$ for a random variable $X$?

For a random variable $ X \geq 0 $ and $E[X^2] < \infty $, I'm asked to prove the following: $$ P(X> 0) \geq \frac{(E[X])^2}{E[X^2]}$$ It makes intuitive sense to me that it must be the case, ...
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351 views

Order Statistics of Poisson Distribution

I have been given the following question, Let $n ≥ 2$, and $X_1, X_2, . . . ,X_n$ be independent and identically distributed $Poisson (λ)$ random variables for some $λ > 0$. Let $X_{(1)} ≤ ...
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1answer
32 views

Getting From Concentration Inequality to Interval Length

I've seen this used some times and I would like to ask what steps are taken on the way to getting there: E.g. assuming bounded variance, we can use Chebyshev concentration inequality: for any $t>0$...
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197 views

General solution of expected value of E(f(X))?

This is maybe a trivial question I came up while solving a few examples and understanding Markov/Chebyshev inequalities and subsequently in evaluating Chernoff bounds. Suppose $X$ is a random variable ...
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74 views

Machine learning for inequalities

This is a very general question about machine learning. Two of the most standard problems in ML are classification and regression. E.g. if we have pictures of buildings, we can classify them as two-...
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136 views

Algebra in Cantelli-Cheybyshev Inequality Proof

I am confused by the following (possibly simple) algebra in the proof of the Cantelli-Cheybyshev inequality. I am following Rohatgi and Saleh (2015, Section 3.4 Lemma 1) where we plug in $\phi(x)=(x+c)...
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98 views

Two distributions, same mean, different variance: Stochastic dominance for deviation from mean?

Say you have two (bounded) random variables, $X$ and $Y$, on the same discrete probability space, such that $E(X)=E(Y)$ but $Var(X) < Var(Y)$. Do I know that, for any $k \geq 0$, $$ \text{Prob}(|X-...
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35 views

Bounds on quantiles of the minimum of summations of (possibly dependent) random variables

Suppose I have $2N$ continuous random variables $X_1, \ldots, X_N, Y_1, \ldots, Y_N$ and that I can evaluate the quantiles of the respective distributions. Given a value $w \in [0, 1]$ I would like to ...
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2answers
45 views

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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29 views

Simplification using Cauchy Schwartz Inequality

Can someone please help me understand how the last step in the highlighted part in the equation is arrived at? I did not get how the Cauchy Schwartz Inequality comes into play here. The full paper is ...
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1answer
64 views

Intuition behind the no convergence of the variance of sum of random variables

$$Var[\bar{X}] = \sigma^2/n $$ $$Var [\sum{X}_i] = n\sigma^2$$ $$lim_{n \to \infty} Var[\bar{X}] = 0 $$ wich means at $\infty$ we will always get the same $\bar{X}$ after every simulation. I ...
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50 views

Effect of scaling on the mean of random variables

Consider two possibly correlated scalar random variables $N$ and $X$. It is known that $1\leq N \leq N_{\max}$. Given that $\mathbb{E}[NX] \leq 0$, does it always hold that $\mathbb{E}[X] \leq 0$? ...
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45 views

How to prove the concentration equality for standard normal?

The following inequality is given in some of Yale's online lecture notes $$P(|Z|>x) \leq 2 \sqrt{2 \pi} \phi(x)$$ Where $Z \sim N(0,1)$ with density $\phi(x)$. They call it a concentration ...

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