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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Inequality relating expected value and tail probability
I am currently working through Scornet2015 - Consistency of Random Forests.
I'm having trouble understanding a specific inequality that is used in the proofs without further explanation. I am assuming ...
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What kind of formal guarantee does a confidence interval provide after an observation?
According to my understanding, $C(X)$ is a random variable defined as follows:
Let $\mathcal{P}$ be a family of distributions (defined by user). Let $\alpha>0$. Let $\theta$ be some parameter of a ...
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Is it true that $\langle X^4\rangle \ge 3 \langle X^2\rangle^2$?
Consider a real random variable $X$ with zero mean. Does the following inequality hold in general?
$$\langle X^4\rangle \ge 3 \langle X^2\rangle^2$$
I'm not sure how to prove this or if a counter-...
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Number of samples for Hoeffding's Bound with Gaussian R.V
I am trying to obtain the required number of sample $n$ for a given confidence interval $\alpha$ and $X_1 ... X_n$ which are Gaussian rv with $\mu$ mean and $\sigma^2$ variance. I know that
\begin{...
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Is kernel density estimation sub-gaussian?
Let $X_1, ..., X_n$ be i.i.d. samples drawn from a pdf $f(x)$ on the real line. The kernel density estimator is defined as follows,
$$\hat{f_n}(x) = \frac{1}{nh}\sum_1^n K(\frac{x-X_k}{h})$$
where $K:\...
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Game of guessing or is it! [closed]
I am trying to learn coding and want to create a simple game. The idea is that there are 20 different animals, and you have 5 rooms, each round 5 out of the 20 animals hide in the room, where behind ...
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Inequality on the moment generating function of a centered random variable which is bounded above
I am stuck on the first part of problem 8.2 of the book "A Probabilistic Theory of Pattern Recognition" by Luc Devroye:
Show that for any $s > 0$, and any random variable $X$ with $\...
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Probability of Roots outside the unit disc
Consider a random polynomial $p(z)=\sum_0^n A_i z^i.$ where $A_i,i=0,12,3,..,n$ are iid uniform variables in the interval $(0,1).$ I want to show that that the probability of the root with min modulus ...
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Deriving and visualizing bounds on the conditional probability given marginal conditionals
Suppose I have a binary outcome $Y$ and two binary covariates $X_1$ and $X_2$ with distribution $P(X_1, X_2)$ which I know. In addition to $P(X_1, X_2)$, I know $P(Y\mid X_1)$ and $P(Y\mid X_2)$. I ...
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probability of being within a 2D grid cell
Consider two joint-normal gaussian variables A and B with known probability density. A and B form a point in 2D space. What is the probability that the sample point falls within a rectangular grid ...
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Proof of inequality $P(X \geq \lambda) \leq \frac{\sigma^2}{\sigma^2 + \lambda^2}$
Hello I have the assumptions that $E(X) = 0$ and $\operatorname{Var}(X) = {\sigma}^{2}$ . Why does this inequality hold $P(X \geq \lambda) \leq \frac{{\sigma}^{2}}{{\sigma}^{2} + {\lambda}^{2}}$ for $ ...
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If all moments of a non-negative random variable X are larger than those of Y, is P(X>x) larger than P(Y>x)?
$X$ and $Y$ are two non-negative continuous random variables. The moments of $X$ are $\mu_i$ while that of $Y$ are $\nu_i$.
We know that $\mu_1=\nu_1$ and $\mu_i \ge \nu_i$ for $i=2,3,\ldots$
Can one ...
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Is it true for rvs $X,Y$ where $E[X]=E[Y]$ & $V[X] \geq V[Y],$ the Jensen gap of $X$ is larger or equal the Jensen gap of $Y?$
Is it true for rvs $X,Y$ where $E[X]=E[Y]$ and $V[X] \geq V[Y],$ the Jensen gap of $X$ is larger or equal the Jensen gap of $Y?$
It seems intuitive. I failed to prove it or find a reference.
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Sum of i.i.d. random variables for which Chebyshev inequalities are tight
Chebyshev's inequalities: Let $X$ be a random variable with finite expected value $\mu$ and finite non-zero variance $\sigma^{2}$. Then for any real number $\delta > 0$,
$$ \Pr[|X - \mu| \geq \...
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Calibrating the probabilities of Ridge Classifier on imbalanced dataset
I have a classification project on an imbalanced dataset (HomeCredit Kaggle dataset) and I have chosen Ridge Classifier (sklearn's implementation) as the most efficient both in terms of time and in ...
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Bounded in Probability
Consider any random variable $X$ with cdf $F_X(x)$. Then given $\epsilon > 0$, we
can bound $X$ in the following way. Because the lower $limit$ of $F_X$ is $0$ and its upper
$limit$ is $1$, we can ...
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Proof of corollary of Hoeffding's inequality
I need to proof a corollary of Hoeffding's inequality, and since I'm not used to doing proofs I really don't know where to begin.
Hoeffding's inequality:
Let $X_1,...,X_n$ be independent real-valued ...
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Asymptotic behaviour of product of normal r.v.s
Let $X \sim N(\mu ,1)$ and $Y \sim N(\mu, 1)$ where we have $\mu >0.$ I'm trying to evaluate asymptotically the tail distribution function of product of these two random variables. Let $x>0$, ...
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Inequality of $f(\mathrm{Mode}[X])$ and $\mathrm{Mode}[f(X)]$?
Say I knew the distribution of a continuous variable $X$. If somebody randomly picked out a single instance $x$ and asked me to bet on its value, I would go with the mode of the known distribution of $...
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Equality in Gaussian Poincare Inequality
The Gaussian Poincare inequality states that: for $f: \mathbb{R}^n \to \mathbb{R}$ and $Z\sim \mathcal{N}(0,I)$, we have that
\begin{align}
Var(f(Z)) \le E[ \| \nabla f(Z)\|^2].
\end{align}
My ...
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Using Chernoff bound as an approximation to upper binomial tail
I'm curious about how well the Chernoff bound approximates the value of the upper tail of a binomial distribution.
It is well known that, for $X\sim B(n,p),\ \delta >0$:
$$P(X \geq (1+\delta)np) \...
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Error in applying Chebyshev's inequality
I'm trying to solve a problem using Chebychev^' s Inequality:
"Suppose that X is a random variable with mean and variance both equal to 20. What can be said about P(0<X<40)?"
P(|X-μ|≥...
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Prove or disprove : $P[A|B] = P[B]$, the A and B are independent? Is this right?
SOrry if this is extremely easy.
I did the following but I'm a little bit unsure about it:
Let $A=B$, and $P[A]>0$.
Then $$P[A|A] = P[A]$$
But A is not independent with itself:
$$P[AA] = P[A] \neq ...
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What does "tensorization" mean in the entropy tensorization inequality?
I am reading high dimensional statistics written by Wainwright. In chapter 3.1.4 tensorization of entropy is used to extend the entropy bound for univariate functions to multivariate cases. As far as ...
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Limit of absolute value of expected value of the sequence of random variables under L^p convergence
I want to prove this property.
If $X_n \longrightarrow X \: in \: \mathbb{L}^p(\Omega)$, then $E[|X_n|^p] \longrightarrow E[|X|^p]$
I tried many ways to do it, including using Minkowski's inequality ...
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Why do we need Jensen inequality for variational autoencoders?
Just to clarify, I think I understand all the derivations in context of VAEs pretty well; however, there is one last thing that I need explained. There are multiple related derivations of the evidence ...
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Solving for the parameter of an exponential distribution
Suppose I have a random variable $X$ where $X$ follows an exponential distribution of the following form:
$$f_X(x) = \frac{1}{\lambda}e^{-\frac{x}{\lambda}}$$.
I want to find the value of $\lambda$ ...
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Product of a series and asequence of random variables
Suppose $\{a_n\}_{n \in N} $ be a bounded sequence positive numberse bounded by $b$ and $Z_j,j=1,2,3,...$ be a sequnce of complex random variables such that $E(|z_j|)<\frac{1}{\sqrt{(j+1)m}}$ for ...
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For $Y \geq 0$, prove that $Pr(Y \geq k) \leq E(Y)/k$
Let $Y$ be a non-negative random variable, $k$ be any positive constant, show that $Pr(Y \geq k) \leq E(Y)/k$.
My attempt (using integration by parts):
\begin{align}
\int_0^k y \,dF(y) &\leq E(Y) \...
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How to prove this inequality?
For any nonnegative random variable $X$ independent of $U$ where $U \sim \operatorname{Uniform}(-t,t)$ and any $t\ge 0,$
$$P(X+U\ge t)\le\frac{E(X)}{2t}.$$
Any hints to prove this inequality?
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Probability Conditioned on Inequality
Assume that $A \sim \mathcal{N}(0, 1)$, $B \sim \mathcal{N}(0, 1)$. I am trying to calculate $P(A \,|\, A < B)$.
For the sake of this problem, we can assume that $A \perp B$, but (for obvious ...
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Is my way of deriving a statistical test from Hoeffding's inequality correct?
I'm trying to deduce from samples of observations from two finite sets of random variables $X_{1}, ..., X_{n}$ and $Y_{1}, ..., Y_{m}$ that the expected values of the average of those random variables ...
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symmetrization in glivenko-cantelli proof
In this proof of the Glivenko-Cantelli theorem, page 2 of these notes, two types of symmetrization are used. The first transforms the sup of the centered empirical cdf $$P(\sup_{z\in\mathbb{R}}|(1/n)\...
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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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Multivariate Chebyshev's inequality with Mahalanobis distance
In Chebyshev's inequality, we can generalize the 68-95-99.7 rule from normal distributions to bound how much density is within a certain number of standard deviations from the mean.
$$
P\big(
\big\...
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Probability inequality
Let $X_1,X_2,...X_n$ are independent random variables such that $P(X_i=v_i)=1/v_i$ and $P(X_i=0)=1-1/v_i$, where $1\leq v_i\leq n+1$ for $i=1,2,...,n$. I want to prove that $P(X_1+X_2+...+X_n\geq n+1)\...
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Proof utilizing Chebyshev's inequality
I'm being asked to show that $P(|X-\mu|\geq t) \leq \beta_4/t^4,$ where $\beta_4=E((X-\mu)^4)$.
I'm familiar with Chebyshev's Inequality, which similarly states that $P(|X-\mu|\geq t) \leq \sigma^2/t^...
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Independent Bernoulli random variables
My professor left us to solve this problem:
Let $\xi_1, \xi_2,...,\xi_n$ be independent Bernoulli random variables defined on a probability space $(\Omega,P(\Omega),\mathbb{P})$ such that:
$\mathbb{P}(...
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Does Cramer's condition imply strong mixing?
In Theorem 1.4 of D. Bosq the Cramer's condition is a prerequisite for the tail bound of sum of dependent variables. The Theorem is as follows:
Let $(X_t,t\in\mathbb{Z})$ be a zero-mean real-valued ...
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Does Chebyshev's inequality sacrifice a little power for simplicity?
I followed the proof in Chapter 2 of Ross Introduction to probability and statistics for Engineers.
As follows;
Chebyshev inequality
For any K > 1 ( for K< 1 still holds for 0<K_1 but trivial ...
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What are the arguments in getting from the theorem of Vapnik & Chervonenkis (1971) to the common form seen in Devroye, Györfi & Lugosi (1996)?
Context.
The theorem below is attributed to "Vapnik, V., and Chervonenkis, A. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability ...
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Markov's Inequality and convergence in probability
Theorem: (Markov's Inequality):
Let x be a non-negative random variable. Then, for all $b >0$
$P[x \geq b] \leq \frac{E(x)}{b}$.
#################
Suppose that $ x_{t}$ is a non-negative random ...
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Corollary of Hoeffding’s Inequality
Question
I am not from a statistics background. I came across the following corollary of Hoeffding’s Inequality and couldn't find the derivation or proof for it. Could anyone please share some ...
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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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Markov Inequality for Sum
I don't have proper knowledge of probability, I was surfing around internet about Markov's inequality, I found a paper on JSTOR titled "The Markov Inequality for Sums of Independent Random ...
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Bounding sum of quartic deviations from sample mean
[Cross-posted here with no answers for a few days]
I came - to the very best of my knowledge from reading the source - across the following statement in The Jackknife and Bootstrap, Shao and Tu, p. 87:...
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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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Proving Chebyshev's Inequality
I'm working on proving Chebyshev's Inequality. I watched this YouTube video and it almost makes sense. There is one step in the proof I don't understand. Using Markov's Inequality you substitute ...
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Can we use Frechet inequalities to infer significance of collections of hypothesis tests?
There are numerous issues that have been identified both in the theory and practice of $p$-values, including the arbitrariness of confidence levels, interpretation, and tail-risk in the ...
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