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0
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0answers
11 views

Express similarity measure in terms of probability

I am trying to express similarity measures between objects in 2 sets. Below are the details of my measures. 1) Compare object - 1 with all the objects in set-2 (object - (2-9)). 2) Similarity measure ...
0
votes
0answers
11 views

Empirical Bernstein Bound for distributions outside range(0,1)

I'm working on using empirical Bernstein bounds to estimate the mean difference between 2 variables from different distribution. The algorithm samples from each variable until it detects a ...
0
votes
0answers
23 views

If distance correlation $DCOR(X,Y) = 0.5$ then are X and Y dependent or independent?

If distance correlation $DCOR(X,Y) = 0.5$ then are $X$ and $Y$ statistically dependent or independent? What about when $DCOR(X,Y) > 0$, are they statistically dependent for sure even when it's less ...
4
votes
1answer
35 views

Binomial and Poisson issues (Jacod and Potter)

I've been reading through Probability Essentials by Jacod and Potter (2nd edition). I'm on a voyage to do every single exercise in the book. The following problems I am unsure of is as such: 5.11) ...
2
votes
1answer
32 views

Expressing conditional probability with inequality in condition

Is there a convenient way to express $$ p(A\leq a |B\leq b \land C\leq c), $$ when all I've got is an expression for $$ p(A\leq a |B = b \land C = c), $$ $$ p(A\leq a |B = b), $$ and $$ ...
2
votes
0answers
45 views

Bound for the variance of 1/X, where X is a Gaussian RV

Consider the following problem: Let $X \sim N(\mu, \sigma^2)$ and assume that $|\mu| \gg \sigma^2$. Then, we construct a new random variable $Y = 1/X$ with pdf $$f_Y (y) = \frac{1}{\sqrt{2 \pi} ...
0
votes
0answers
6 views

Gaussian approximation for general Covariance matrices

There are quite a few results in the literature that provide Berry-Esseen type bounds for the convergence of a standardized sum $S_n:=\frac{1}{\sqrt{n}} \sum_{i=1}^n X_i$ of iid random vectors in ...
4
votes
0answers
43 views

Probability bound that the sum of random variables with positive mean is not positive

EDIT: The distribution I gave below is not necessarily correct for $n > 1$, so the bound I gave below does not hold. Let $X_1,\dots,X_n$ be independent and identically distributed random ...
3
votes
3answers
65 views

Inequality of binomial probabilities

I need to show the following: I have two binomial random variables $X \sim BIN(m,p_1)$ and $Y \sim BIN(m,p_2)$, where $p_2 \geq p_1$. I want to show for any fixed constant $c \in \{0,...,m-1\}$ that ...
1
vote
0answers
26 views

Relation between expectations of two random variables

I have two random variable X and Y. I know that $$E_X\left[X \log \frac{X}{e}\right] < E_Y\left[Y \log \frac{Y}{e}\right]$$ Using the above relation can I say anything about the relation between ...
-2
votes
2answers
55 views

$\mathbb{P}(A_1)≤\mathbb{P}(A_1)$ in Boole's inequality ($n=1$) proof?

Why does this proof use $≤$ in the $n=1$ (induction base case) case for Boole's inequality, when in fact it's an equality? That is, why claim, $\mathbb{P}(A_1)≤\mathbb{P}(A_1)$, when it should be a ...
1
vote
0answers
36 views

Finding Probability Distribution Parameters Satisfying a Particular Condition [closed]

Condition: $$E[\text{max}(X,Y)] \leq E[\text{max}(K,Y)]$$ Here, $X,Y$ are random variables. $K$ is a constant. The distribution for $Y$ is known. Question one: Is it possible to find the ...
0
votes
0answers
27 views

Related to Chebychef's inequality

Please help me with this problem. Suppose that $X$ is a random variable for which $E(X)=\mu$. Prove that $$\Bbb{P}(|X-\mu|\ge t)\le \frac{E[(X-\mu)^4]}{t^4}$$ The only thing I have been able to do is ...
9
votes
0answers
140 views

Special probability distribution

If $p(x)$ is a probability distribution with non-zero values on $[0,+\infty)$, for what type(s) of $p(x)$ there exist a constant $c>0$ such that $\int_0^{\infty}p(x)\log{\frac{ ...
2
votes
0answers
70 views

Prove $\int_E |f| d\mu < \infty$, $\lim \int_E f_n d\mu \to \int_E f d\mu$

Suppose we have a measure space $(\Omega, \Sigma, \mu) = (\mathbb R, \mathscr B(\mathbb R), \lambda)$ (Alternatively, we can consider a similar probability space on some subset of $\mathbb R$ that has ...
0
votes
0answers
32 views

Expectation of maximum of a sequence of identically distributed but not independent RVs

Let $X_1, X_2, \dots, X_n$ be identically distributed but not necessarily independent random variables with $E|X_i|^{\alpha} < \infty, \alpha >1$. In part a) we are required to show that for ...
0
votes
0answers
20 views

About calculating the marginal PDFs from a joint PDF [duplicate]

I am trying to find the marginal distributions of a given joint probability density function. The joint density is $f(x,y) = xe^{-x(y+1)}$ for $x$ and $y$ positive and zero everywhere else. If I have ...
2
votes
2answers
99 views

Prove $E[|Z|] < \infty \to E[|Z_n|] < \infty$

Let $Z$ be an integrable random variable on filtered probability space $(\Omega , \mathscr F, (\mathscr {F_n})_{\{n \in \mathbb{N}\}}, \mathbb P)$ Define $Z_{n} := E[Z|\mathscr {F_n}]$. Show that ...
0
votes
0answers
27 views

Upper bound on the expected trace of the inverse of a sum of psd matrices

Let $X$ be an $n\times n$ rank-deficient positive semi-definite matrix and let $Y$ be an $n\times n$ full-rank, positive definite matrix. Suppose that $\mathbb E \ \text{tr}(X)$ and $\mathbb E \ ...
1
vote
1answer
45 views

Distribution of a transformation of a random variable

I have started off by this: $F_Y(Y)=P(Y\leq y)=P(X^2 \leq y).$ Now, I have been told that $P(X^2 \leq y) = P(- \sqrt y \leq X \leq \sqrt y) $. I don't quite understand why this is and any help ...
1
vote
0answers
36 views

Better Moment Inequalities

How do you determine if a moment inequality is better than another? Say for example, compare the Chebyshev's inequality with this nameless inequality where P{|X|≥ Kσ} ≤ (μ4-σ4)/(μ4+K4σ4-2K2σ4). I ...
2
votes
1answer
79 views

convergence rate of Pearson correlation

I am looking to find the finest bound possible for $$\mathbb{P}\left( | \hat{\rho}(X,Y) - \rho(X,Y) | \geq t\right) \leq ?$$ where $\rho$ is the Pearson correlation defined as $$\rho(X,Y) = ...
1
vote
0answers
102 views

Using Jensen's Inequality to identify blackbox function as convex?

** Edits: In light of the comments I received: I'll appreciate any elaboration on the questions I present. If you can provide examples of what I would need (assumptions, test cases, etc), ...
2
votes
0answers
22 views

Properties of the KL topology [reference request]

I'm trying to understand better what are the implications of a sequence of random variables $X_n$ converging toward some limit $X$ in the KL topology, ie the probability density functions are such ...
0
votes
0answers
16 views

Concentration inequalities for product of gaussians

Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $n$ gaussian random variables with zero means and equal ...
0
votes
0answers
47 views

Application of probability-inequalities in machine learning. Hoeffding's inequality

I would like to know a illustrative example of Hoeffding's inequality in machine learning? Does it have something to do with confidence intervals? Thanks in advance!
0
votes
1answer
252 views

How to use Hoeffding's Inequality to find a confidence Interval?

I want an example that shows how to use Hoeffding's inequality to find a confidence interval for a binomial parameter p (probability of succes). Thanks in advance!.
5
votes
1answer
92 views

Comparison of Bernstain and Chebyshev inequalities applied to Bernoulli distribution - simulation in R gives unexpected results

I'm trying to compare Bernstein and Chebyshev inequalities applied to Bernoulli distribution with parameter $p$. More specifically - how good are bounds they give for different sample sizes. I wrote ...
9
votes
1answer
168 views

Does convex ordering imply right tail dominance?

Given two continuous distributions $\mathcal{F}_X$ and $\mathcal{F}_Y$, It is not clear to me whether the relation of convex dominance among them: $$(0)\quad \mathcal{F}_X <_c \mathcal{F}_Y$$ ...
2
votes
1answer
59 views

Shannon entropy and inequality of expectations

Consider two distinct probability distributions $P(X)$ and $Q(Y)$---defined on the same domain---with (Shannon) entropy of $H(X)$ and $H(Y)$. I am interested to prove that $$ H(X) \leq H(Y) \implies ...
2
votes
0answers
63 views

Dvoretzky–Kiefer–Wolfowitz inequality hold for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
2
votes
0answers
50 views

bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \stackrel{iid}{\sim} P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one ...
5
votes
1answer
144 views

Berry-Esseen Theorem with Continuity Correction

Given independent but non-identical random variables $X_1, X_2, \ldots ,X_n$ with $E[X_i]=0,$ $E[X_i^2]=\sigma_i^2=1$ and finite absolute third moments $\rho_i=E[|X_i|^3].$ Let $$S_n = {\sum_{i=1}^n ...
1
vote
1answer
32 views

Inequalities for maxima over random functions

Let R be the set of real numbers. Say we have a function $f(x,X) \in R $ where $X \in R^d$ is a random variable over $\Omega$ and $x \in R$. I'm searching for an upper bound for the expected value of ...
11
votes
3answers
205 views

Regarding convergence in probability

Let $\{X_n\}_{n\geq 1}$ be a sequence of random variables s.t $X_n \to a$ in probability, where $a>0$ is a fixed constant. I'm trying to show the following: $$\sqrt{X_n} \to \sqrt{a}$$ and ...
2
votes
0answers
30 views

upper bound for expected maximum of difference of two kernel-Estimations

I'm searching for an upper bound for a function like $$ E\left[ \max_{x \in R} \left( \frac{ \sum_{i=1}^n K(\frac{x-X_i}{f(x,X_1, \dots X_n)}) \cdot Y_i } { \sum_{i=1}^n ...
2
votes
2answers
121 views

Moving an expectation inside a probability?

Let $X$ and $Y$ be two independent random vectors such that $E[X^TY]>0$, and all components of $X$ are positive valued. Then, is it true that, $P_{X,Y}\{X^T Y > 0\} \le P_{Y}\{E_X[X]^T Y > ...
8
votes
0answers
184 views

A question related to Borel-Cantelli Lemma

Note: Borel-Cantelli Lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ $$\sum_{n=1}^\infty P(A_n) =\infty \textrm{ and } ...
1
vote
1answer
41 views

How to combine probability inequalities that are w.r.t. different random variables?

Let $x$ and $z$ be two independent random variables. Suppose I know the following two facts: $P_z[f(x,z) < g(x)] > 1-\delta$ uniformly for all x; $P_x[g(x) < h(x)] > 1-\delta_h$ How can ...
2
votes
0answers
60 views

Probability inequality implication

Let $X$ and $Y$ be independent random variables, and $A(X,Y)$ a predicate with values true or false. Suppose we know the following: $P_Y[\forall X, A(X,Y) \Rightarrow B(X)] > 1-\delta$. Can we ...
4
votes
2answers
124 views

$E(X)E(1/X) \leq (a + b)^2 / 4ab$

I've worked on the following problem and have a solution (included below), but I would like to know if there are any other solutions to this problem, especially more elegant solutions that apply well ...
1
vote
1answer
161 views

Chi-square test on a sample from population having an unequal distribution

I got a sample with an unequal gender distribution ($30$ males to $200$ female). This distribution is quite "representative" for the population the sample is taken from. Nevertheless, in a paper, ...
1
vote
0answers
36 views

Results stronger than Dvoretzky–Kiefer–Wolfowitz inequality?

There is the DKW inequality which controls the extent to which the empirical cdf of a sample from a real-valued random variable differs from the true cdf. Are there any stronger results (i.e. ...
2
votes
0answers
35 views

Is this a true inequality over multinomial parameters?

Let $p_1,\ldots,p_k$ be the multinomial parameters sampled from a Dirichlet distribution. Assume $\bar{p_1},\ldots,\bar{p}_k$ be the mean of Dirichlet distribution. Then, $$Pr( ...
2
votes
0answers
22 views

Identifying values responsible for producing inequality or concentration within a distribution

(This may be a dumb question I simply don't know enough to know is dumb, but it's foxing a surprisingly large chunk of my social circle) I have a dataset web requests - specifically, the page, and ...
1
vote
0answers
40 views

Tail bound on dependent sum

Let R be a random variable, and let X_i be a nonlinear, nonconvex function that takes R and outputs a value in [-1,1]. I'm trying to get a tail bound on the following dependent sum: ...
3
votes
1answer
91 views

Understanding a step of a proof of Markov's inequality

$X$ is a positive valued random variable, $\theta$ is greater than $1$, $P$ is a probability measure. Why is the following inequality true: $$\int_{\theta E(X)}^{+\infty} x dP_{X}(x) \ge ...
1
vote
0answers
29 views

Families of distributions with bounded variations

For what family of probability distributions $f(x)$ do we have the following property? $$ \forall x, \quad \int f(u) - f(u+x) du \leq L\| x \| $$ for some $L$. Can we say anything about the ...
1
vote
1answer
244 views

Tail bounds for Beta distribution

Say $X\sim\mathrm{Beta}(\alpha,\beta)$. Are there any "nice" closed form upper bounds for the tail probability $P(X\geq\epsilon)$, that are reasonably tight when $\beta$ is large? By "nice" I mean ...
0
votes
0answers
32 views

how can Cauchy-Schwarz help here?

From http://math.stackexchange.com/questions/1004544/if-mathrme-x2-exists-then-mathrme-x-also-exists If $\mathrm{E} |X|^2$ is finite, then $\mathrm{E} X$ exists and is finite, because $$ ...