Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
1answer
41 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-...
3
votes
1answer
32 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)...
1
vote
2answers
40 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-...
1
vote
0answers
18 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 ...
2
votes
2answers
37 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 ...
1
vote
0answers
24 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 ...
1
vote
1answer
44 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 ...
3
votes
1answer
32 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 $\mathbb{E}[X] \leq 0$? ...
0
votes
1answer
37 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 ...
1
vote
0answers
39 views

Tight upper bound on the expectation of a concave function

N is a random variable whose sample space is [0,$\infty$). I have an expression in terms of the expectation of this variable and I want to find a tight upper bound on the whole expression. The ...
0
votes
0answers
46 views

Establishing an upper bound for the tail probability $P(X-\lambda \geq z)$ for any $z>0$, where $X$ is Poisson r.v. w/ parameter $\lambda$

Poisson random variable $X$ with the parameter $\lambda$ has, respectively, the pmf and the moment generating function of the forms $$P(X = k) = \dfrac{e^{-\lambda}\lambda^k}{k!}, \quad k=0,1,2,\...
4
votes
1answer
39 views

Comparing suprema of inner products of Gaussian variables

I'm given two i.i.d. standard normal vectors $x, y \sim \mathcal{N}(0, I_n)$, and vectors $a \in \mathbb{S}^{n-1}$, the unit sphere in $n$ dimensions. Additionally, given a set $S \subseteq [n]$, I ...
2
votes
1answer
77 views

An inequality giving a sharper bound than that given by the Chebyshev's?

Let $X > 0$ be a random variable; let $P$ be the underlying probability measure; let $\delta > 0$. I wonder if there is already in probability literature a known result giving a sharper bound ...
3
votes
0answers
92 views

Which concentration inequalities apply when moments are infinite?

I have 2 questions: Suppose I have a finite mean but an infinite variance for a discrete distribution w/support $\{1,2,\dots\}$. Is there any probability inequality tighter than Markov in this case? ...
1
vote
1answer
38 views

Concentration for Conditional Random Variable

Consider a conditional random variable \begin{equation} X = \begin{cases} Y & \quad\quad ,X \in A \\ Z & \quad\quad ,X \in A^\complement \end{cases} \end{equation} $Y$ ...
5
votes
1answer
78 views

Upper bound for the probability $P\left[\left|\frac{Y_n}{n}-p^2\right|>\varepsilon\right]$

Let $X_1,X_2,\cdots,X_{n+1}$ be independent random variables with $$P(X_i=1)=p=1-P(X_i=0)\quad\text{ for all }i$$ Define $Y_i$ to be the number of $i$'s such that $X_i=X_{i+1}=1\,,\quad i=1,2,\...
0
votes
0answers
77 views

Why is it that generalization is not guaranteed for 1-nearest neighbors?

I wanted to understand from a statistical learning theory perspective why 1-nearest neighbors doesn't have generalization. I define generalization as empirical risk converging to expected risk as N ...
1
vote
0answers
44 views

Conditional distribution relations

There is a probability density function of the form, $f_S(s)=\displaystyle\iint f_S(s|x,y)f_{X,Y}(x,y)dxdy$ that is used for evaluation of expectation of some monotonic function $\mathbb{E}[g(S)]=\...
2
votes
0answers
63 views

Lower bounds on covering numbers for sparse vectors

Consider the set $S_k$, which is defined as the subset of $k$-sparse vectors in the unit sphere in $d$ dimensions: $$ S_k \triangleq \left\{ x \in \mathbb{R}^d : \| x \|_2 = 1, \, \left|\operatorname{...
5
votes
1answer
106 views

How good an approximation is sampling with replacement to sampling without replacement?

I'm learning about probability with Feller's book and he states that, when the population size $n$ is big in comparison with the sample size $r$, then $n_r$, which is a shorthand for $\frac{ n!}{(n-r)!...
5
votes
1answer
97 views

Covering the unit sphere with sparse vectors

I'm looking for references for covering the $d$-dimensional unit sphere $$ \mathbb{S}^{d - 1} = \left\{ x \in \mathbb{R}^d : \| x \| = 1 \right\} $$ I'm trying to cover $\mathbb{S}^{d-1}$ with ...
1
vote
1answer
21 views

Concentration Square Increments of a MDS

I have a martingale difference sequence $\{ X_t \}$ where each $X_t$ is subGaussian. Are there concentration inequalities for $$ \sum_{t=1}^T X^2_t - E \left( \sum_{t=1}^T X^2_t \right) $$
0
votes
1answer
50 views

Is my solution correct for this measure-concentration related task?

I'm reading the book "Concentration inequalities" by Boucheron, Lugosi, Massart. There is an exercise section after each chapter. I've tried to solve one and would like to understand, whether it would ...
3
votes
1answer
102 views

Accuracy of empirical cumulative distribution function

I have a random variable with an unknown distribution and I want to find its cumulative distribution function. I sample the distribution $N$ times, with $$X_1, \dots, X_N$$ being iid random ...
1
vote
1answer
57 views

$CTE(p)$ is generally greater than $VaR(p+\frac{1}{2}\cdot(100-p))$, $p$ being a percentile

Let's assume we are in the insurance business and the values we are observing are losses. So there is a general statement that says the Conditional Tail Expectation at percentile $p$ is usually ...
12
votes
1answer
692 views

Oracle Inequality : In basic terms

I'm going through a paper that uses oracle inequality to prove something but I'm unable to understand what it is even trying to do. When I searched online about 'Oracle Inequality', some sources ...
2
votes
0answers
72 views

Tight bound for Binomial distribution or, equivalently, the Incomplete Beta function?

Suppose $X \sim Binomial(n,p)$ with known $n$ but unknown $p$, and let $G(p,k) = P[X \geq k)$ for $k=0, \ldots, n$. I am looking for a tight upper bound on $|G(p_1, k) - G(p_2, k)|$ for some given $k$....
3
votes
0answers
42 views

Concentration inequalities for weighted sums of gaussians

Suppose that $x \sim \cal{N}(0,I_d)$ be a $d$-dimensional standard Gaussian vector and let $x_1,\ldots,x_n$ denote $n$ i.i.d. samples drawn from the same distribution. For some fixed vector $\theta \...
0
votes
0answers
22 views

Expectation of a product, knowing partial information

Suppose $f$ and $g$ are positive valued functions of a random matrix $X$, and we know $f(X) \le 1$ w.p. $1-\delta$. We also know $E_X[g(X)] = c < \infty$. Can we upper bound the following ...
3
votes
0answers
328 views

Concentration of maximum of subexponential random variables

I'm looking for a concentration bound on the maximum of a collection of sub-exponential random variables, which are not necessarily independent. More specifically, I have the following collection: \...
0
votes
1answer
119 views

the normalization constant

On page 4 of this article, the authors wants to find the normalizing constant $c$ but it is very hard to compute so I used the same formula to bound $c$. Take the non-negative integers $~0 \leq x_1,...
1
vote
0answers
76 views

Cantelli Inequality Variant

In the JASA (1968, vol. 63, no. 324, pp. 1522-1525) article "How Deviant Can You Be?" Paul Samuelson notes two versions of Chebyshev's inequality: $$P \left( \lvert {X-\mu} \rvert \geq k \sigma \...
6
votes
1answer
106 views

Zero-mean RV $X$, probability of being positive using moments

For zero-mean RV $X$ with finite fourth moment, prove that $$ P(X>0)\ge \frac{\mathbb{E}(X^2)^2}{4\mathbb{E}(X^4)} $$ I tried Chebyshev with adding $t$ to both sides, but I could not get fourth ...
1
vote
0answers
37 views

Convergence of a sequence in finite number of steps

Here is the setup of my problem. It is a sequential problem and there are two possible actions A and B. Now, when either action $A$ or $B$ is taken at the $j$th time point, we observe some outcome say ...
3
votes
0answers
58 views

Characteristic function inequality

Random variable $X$ and its characteristic function $\phi_X(t)$ then $$\Pr\left(|X|>\frac2T\right) \leq 2\left(1 - \frac1{2T}\int_{-T}^{T}\phi_X(t)dt\right) $$ I cannot find a way how to ...
2
votes
1answer
73 views

Why does conditional expectation have this property for independent random variables?

For a reference, please see pp. 53-54 of Boucheron, Lugosi, Massart, Concentration Inequalities: A Nonasymptotic Theory of Independence. Let $f: \mathcal{X}^n \to \mathbb{R}$ be a measurable function (...
0
votes
1answer
36 views

Increased probability of event during period of time

In game of FIFA there are packs by opening which a user receives soccer player cards. The higher the rating of a player card the rarer it drops depending on some kind of random number generator. Since ...
0
votes
0answers
109 views

Linear combination of truncated normals

I am trying to calculate the following expression: $$ Z = \mathbb{E}\left[\left| \langle \mathbf{a}, u \rangle \right| \right] = \left| \sum_{i=1}^d a_i u_i \right|, \quad \left\| u \right\| = 1 $$ ...
4
votes
2answers
831 views

Using Chebyshev's inequality to obtain lower bounds

Let $X_1$ and $X_2$ be i.i.d. continuous random variables with pdf $f(x) = 6x(1-x), 0<x<1$ and $0$, otherwise. Using Chebyshev's inequality, find the lower bound of $P\left(|X_1 + X_2-1| \le\...
1
vote
0answers
45 views

Bounds on tail conditional expectation of random variable given variance

Given a random variable $X$ with CDF $F(X)$, mean $E(X)=0$, and variance $Var(X) =\sigma^2$, I would like to bound the tail conditional expectation where $X$ is in the tail with probability $1-p$: $E(...
4
votes
1answer
153 views

Proof that $E(|X_1 - X_2|)$ is bound by twice the mean

Let $X_1$, $X_2$ be iid random variables. How do I show that for non-negative variables $E(|X_1 - X_2|)$ is bound from above by twice the expected value of $X_1$ (or $X_2$)?
6
votes
1answer
235 views

How do I show this using the Cauchy-Schwarz inequality

From Two Population We have two $p$ dimensional sampels $\mathbf{X}_1$ and $\mathbf{X}_2$ with sample size $n_1$ and $n_2$ respectively, Let We have $\mathbf{\Lambda} = \mathrm{diag}\{(\sigma_{11}^2 + ...
1
vote
1answer
202 views

Does this Bonferroni styled inequality also hold for characteristic functions?

This is the popular Bonferroni inequality. Does it also hold for characteristic functions of random variables, as in when $P(A_i)$ is replaced by the characteristic function $\chi(A_i)$ and so forth? ...
8
votes
3answers
300 views

What's higher, $E(X^2)^3$ or $E(X^3)^2$

So I had a probability test and I couldn't really answer this question. It just asked something like this: "Considering that $X$ is a random variable, $X$ $\geqslant$ $0$, use the correct inequality ...
1
vote
1answer
1k views

Kullback-Leibler divergence lower bound

Are there any (nontrivial) lower bounds on the KL-divergence between two densities? Informally, I am trying to study problems where $f$ is some target density, and I want to show that if $g$ is chosen ...
4
votes
0answers
95 views

Uniform convergence of expectation?

Let $U_i$, $i = 1 ,2\dots, $ be i.i.d. standard normal random variables. For every $n$, let $f_n:\Theta \times \mathbb R^{n} \to [0, 1]$, where $\Theta$ is some compact subset of $\mathbb R$ [may ...
1
vote
1answer
335 views

Bounds on quantiles of the sum of (possibly dependent) random variables

Suppose I have two continuous random variables $X$ and $Y$, and I can evaluate the quantiles of these distributions individually. I aim interested in what kinds of constraints can be put on quantiles ...
1
vote
1answer
94 views

a question about conditional probability inequality

If P(A|E) > P(B|E) and P(A|not E) > P(B|not E), Then is P(A) > P(B) true? If it is, how to prove it? Intuitively, I'm thinking it should be true but I don't know how to prove it.
2
votes
2answers
100 views

A question about conditional probability

Assume $X$ is discrete random variable. It has some distribution on integers $0,1,...,m$ Then is $P(X>k | X=m) \ge P(X>k)$ true? If it is, how to prove it in a rigorous way? For me, it is ...
1
vote
1answer
179 views

Chebyshev Inequality Homework Question

A number of lightbulbs have lifetimes X that are iid and exponentially distributed with a mean of 1/4. As a lightbulb fails, it is replaced with another until the bulbs run out. Using Chebyshev ...