2
votes
0answers
22 views

Inequality similar to Hoeffding

I have a coin with heads probability $p_1$. I toss it $n_1$ times. Let $\hat{p}_1$ be the empirical heads probability. Then we know from Hoeffding that $$P\left( \left|\hat{p}_1-p_1 \right| \geq ...
0
votes
2answers
47 views

Inequality with moments

Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here). Is $m(f^3) \le m(f^2) m(f)$?
3
votes
1answer
30 views

Motivation behind steps in proof of Hoeffding Inequality

The lemma that is proved for proving Hoeffding's inequality is: If $a\leq X\leq b$ and $E[X]=0$, $E[e^{tX}] \leq e^{\frac{t^2(b-a)^2}{8}}$ Here's a link to the proof: ...
2
votes
1answer
53 views

Bell's inequality

Let $\xi, \eta, \zeta$ be random variables such that $|\xi|, |\eta|, |\zeta| \le 1$. I need to prove such inequality: $|\mathbb{E}(\zeta \xi)-\mathbb{E}(\zeta \eta)| \le 1 - \mathbb{E}(\xi \eta)$ ...
0
votes
1answer
20 views

why does this inequality hold with expectations of supremums

I'm reading a proof on criterion for a class to be Glivenko-Cantelli and I don't see why this holds? $$E \sup_{g\in G} \left|E\left[ \frac{1}{n}\sum_{i=1}^n(g(X'_i)-g(X_i))\big|X_1^n\right]\right| ...
2
votes
1answer
55 views

Proving the Kochen-Stone lemma using the Paley-Zygmund inequality

I am trying to understand a proof to a lemma by Kochen and Stone which appears here, using the Paley-Zygmund inequality. I'll repeat the proof in a detailed manner, and explain what bothers me about ...
0
votes
0answers
25 views

Is there any version of Jensen's inequality for quasiconvex function

I am looking for some generalization of Jensen's inequality for functions $g:\mathbb{R}^n \rightarrow \mathbb{R}$ where $g(x)$ is quasiconvex (or not convex). We known that for convex functions, ...
1
vote
0answers
29 views

Proving that $ (\mathbb E [X^n])^{1/n}\leq (\mathbb E [X^m])^{1/m}$ for $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$

How to prove that for a positive essentially bounded random variable $X$ satisfing $1\leq X\leq 2 \ \mathbb P \text{-a.e.}$ and for any $m,n \in \mathbb N^*$ with $m\geq n$ we have $$ (\mathbb E ...
0
votes
1answer
32 views

Is it true $({\Bbb E|X|^p})^{\frac{1}{p}} \leq ({\Bbb E|X|^q})^{\frac{1}{q}}$, if $p \leq q$?

I ran into this post which shows $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$, $p \leq q$. So I guess it's true $({\Bbb E|X|^p})^{\frac{1}{p}} \leq ({\Bbb E|X|^q})^{\frac{1}{q}}$, if $p \leq q$, ...
0
votes
1answer
41 views

Why $X \geq 0$ and $\Bbb E{X} < + \infty$ implies that $\lim_{y \to 0^{+}}y \Bbb{E}{(\frac{1}{X} | X > y)} = 0?$

Why $X \geq 0$ and $\Bbb E{X} < + \infty$ implies that $\lim_{y \to 0^{+}}y \Bbb{E}{(\frac{1}{X} | X > y)} = 0?$ I'm thinking about replacing $\Bbb{E}{(\frac{1}{X} | X > y)}$ with ...
2
votes
1answer
36 views

Prove that $P[X>\epsilon] \leq M(t)/e^{\epsilon t}$

Prove that $P[X>\epsilon] \leq \dfrac{M(t)}{e^{\epsilon t}}$ Looks like Markov's inequality, it's very easy to derive for $t>0$ $P[X>\epsilon] =P[Xt>\epsilon t]=[e^{Xt}>e^{\epsilon ...
2
votes
2answers
47 views

Proof of Markov's inequality using alternate form of expectation

For nonnegative random variables $X$, there is an alternate expression for the expectation: $$E[X] = \int_0^\infty P(X \ge t) \mathop{dt}.$$ I am familiar with proofs of Markov's inequality $$P(X \ge ...
0
votes
2answers
48 views

Markov's Inequality, only non-negative random variables

I have a question about a Markov's inequality, which states following. Let $X : \Omega \rightarrow \mathbb{R}$ be a non-negative random variable on probability space $(\Omega, \mathscr{A}, P)$ and ...
1
vote
0answers
25 views

Showing an inequality relating two Poisson tail-probabilities

In my research, I've discovered that a property that I am interested in is equivalent to an inequality involving two tail-probabilities of the Poisson distribution. I belive this inequality to be ...
1
vote
1answer
41 views

Lower bound functional binomial r.v.

I am trying to find a bound of the type $\mathbb{E}(|B-\frac{N}{2}|) \geq C \sqrt{N}$ Where $B$ is a binomial variable with parameters $(N,\frac{1}{2})$. The bound doesn't need to be very tight in ...
0
votes
1answer
44 views

Convexity of Binomial Term

I am reading a book on the probabilistic method, and the following claim was made: $\dbinom{y}{n}$ is convex. Why is this the case?
2
votes
1answer
25 views

Bound on the $Q$ function related to Chernoff bound

For the function $Q(x) := \mathbb{P}(Z>x)$ where $Z \sim \mathcal{N}(0,1)$ \begin{align} Q(x) = \int_{x}^\infty \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2} \right) \text{d}u, \end{align} for ...
0
votes
1answer
95 views

Q function and the Chernoff bound

How do we use the Chernoff bound to prove that $$ Q(x)\leq e^{-\frac{x^{2}}{2}} $$ where Q(x) is the probability that a standard normal random variable X takes a value greater than x
1
vote
0answers
33 views

Prove or disprove an inequality involving statistics

Do we have any result in statistics like this: $$|\overline x - \mu_e| \leq \sigma$$ Here $\overline x$ denotes the usual mean of some given discrete observations, $\mu_e$ their median and ...
0
votes
2answers
38 views

Probability inequality proof

I'm stuck on a homework question and don't even know where to start. Here it goes: If A and B are two events which are not impossible, prove that $$P(A\land B)\times P(A\lor B)\le P(A)\times P(B)$$
1
vote
0answers
62 views

Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
0
votes
1answer
52 views

Inequality with monotone functions on power set

Consider a discrete probability space $\left( S, F, P\right)$, where $S = \{ 1, 2, \ldots, N \}$. Consider the set $$S' := \mathcal{P}(S) \setminus \{ \varnothing\} = \{ \{ 1\}, \{ 2\}, \ldots, ...
0
votes
1answer
46 views

Which different probabilistic bounds/inequalities apply when we are given a lower bound on the sample size

Let m be the sample size and $X_i$ be a r.v. that we sample and define a new r.v. such that: $$M_m=\frac{1}{m}\sum^m_{i=1}{X_i}$$ My question is, what type of probabilistic inequalities require some ...
1
vote
0answers
80 views

How to use Chebyshev's inequality or the law of large numbers to a probability question

Let x be a random bit string that takes values $\{1,0\}^n$. Let r be the value of the most significant (MSB) bit of x (and r is a r.v. 1 or 0 that are equally likely). Let g be our guess for the MSB ...
1
vote
0answers
109 views

Cauchy Schwarz inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
1
vote
0answers
59 views

An inequalities on order statistics

Let $X=(X_1,...,X_k)$ be a square integrable random vector in $\mathbb R^k$ and $X_{(j)}$ the $j$-th ordered value of $X$, i.e., $X_{(1)} \leq X_{(2)} \leq ... \leq X_{(k)}$. Prove (or disprove) that ...
3
votes
1answer
59 views

Azuma's inequality: Expected sum of differences

I am looking for an extension of Azuma's inequality which involves the expected sum of squared differences. In particular, recall that Azuma's inequality states \begin{align*} \Pr[X_n-X_0 \geq a] \leq ...
2
votes
2answers
100 views

Inequality for Expected Value of Product

Let $(\Omega, \mathbb{P}, \mathcal{F})$ be a probability space, and let $\mathbb{E}$ denote the expected value operator. Consider the random variables $f: \Omega \rightarrow \{0,1,2\}$ and $g: \Omega ...
4
votes
1answer
40 views

How prove $E|x|^p<+\infty,E|y|^p<+\infty$ ,if $E|x+y|^p<+\infty$

let two random variables X and Y are independent of each other,for some $p>0$,we have $$E|x+y|^p<+\infty$$ show that $$E|x|^p<+\infty,E|y|^p<+\infty$$ my try: I know Minkowski inequality ...
1
vote
0answers
36 views

upper bound on third moment [duplicate]

If I know that $EX^4=1$ and $EX\leq0$, how do I get an upper bound for $EX^3$? I know that by Jensen's inequality, $EX^3\leq1$, but I need the upper bound less than 1.
1
vote
1answer
91 views

Is there any direct relation among first, third, fourth order moments?

Suppose that $X$ is a real-valued random variable, with $EX^4=1$ and $EX\leq0$. Find an explicit constant $c<1$ such that $EX^3<c$.
0
votes
2answers
92 views

Bound on third moment

I'd like to show that if $\mathsf{E}(Z^4)=1$, then $\mathsf{E}(Z^3)\leqslant 1$. I've been trying to use Jensen's inequality to show this, but haven't managed.
0
votes
0answers
29 views

Markov inequality conditions

I'd like to show that $Z-E(Z|Z \leq z)\geq 0$ in order to apply Markov's inequality to $P[Z-E(Z|Z\leq z)>z-E(Z|Z\leq z)]$. Can this be shown given $Z\geq 0$ and $0<P(Z \leq z)< 1$?
0
votes
0answers
32 views

Concentration Inequalities For Matrices? (Around Mean)

I need a result which talks about concentration of a 'random matrix' around its expected value. I need the following: $Pr(||X-E[X]|| \le \epsilon) \ge \hspace{2pt} ? \hspace{8pt}, X \in \mathbb{R}^{n ...
1
vote
2answers
72 views

How to use Chebyshev Inequality

Use Chebyshev Inequality to estimate the probability that in any one day of a business that earns a mean of 100 dollars a day with a standard deviation of 28.87 dollars, that business will make either ...
5
votes
1answer
82 views

Hoeffding inequality adapted to discrete random variables

Given $n$ (real-valued) random variables $X_1, X_2, ..., X_n \in [0, B]$, it can be derived from Hoeffding's Inequality that: $$\mathbb{P}^n\left[ \bar{X} - \mathbb{E}_n[ \bar{X} ] \geq t \right] \leq ...
1
vote
0answers
45 views

A probability inequality

Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume $X_1,X_2,Y_1,Y_2$ are four random variables. Assume $X_1$ and $X_2$ are independent. Is it necessarily true that ...
1
vote
1answer
65 views

$\int_{t=-\infty}^x (G(t)-F(t))\mbox{d}t\geq 0\forall x$ and $\frac{\mbox{d}F(t)}{\mbox{d}G(t)}$ increasing $\Longrightarrow G(x)\geq F(x)\forall x$?

As in the title I wonder if the relation at the left side implies the one at the right side. Better to rewrite it clearly Given are: $\mbox{Info} (1)\rightarrow$ $\int_{t=-\infty}^x ...
2
votes
0answers
42 views

Upper bound for tail of binomial expansion

Let $P,R,T$ be integer constants with $PR$ much greater than $T$. Suppose I flip a coin $PR$ times, each time (independent of other times) getting heads with probability $1/P$. The probability that I ...
3
votes
0answers
51 views

Is Hoeffding's bound tight in any way?

The inequality: $$\Pr(\overline X - \mathrm{E}[\overline X] \geq t) \leq \exp \left( - \frac{2n^2t^2}{\sum_{i=1}^n (b_i - a_i)^2} \right) $$ Is this bound (or any other form of hoeffding) tight in ...
0
votes
0answers
30 views

Poisson probability inequality

How do I find $S$ such that the inequality $CL \le e^{-n\lambda R}\sum_{k=0}^S \frac{(n \lambda R)^k}{k!}$ holds, where $n$ is a positive integer representing number of units in service, $\lambda$ is ...
0
votes
1answer
31 views

Average of m weights equals the average of expected value

I'm reading a paper that says $max\{w_1,\sum_jM^j/m\} = max\{w_1,\sum_i w_i/m\}$. There are $n$ weights $w_1 \dots w_n$ and $w_1$ is the max of the weights. $M^j$ is defined as $\sum_i p^j_iw_i$. ...
0
votes
0answers
46 views

Hoeffding Inequality

Assume that you have $1000$ fair coins labelled as $C_1,C_2,...C_{1000}$. You flip each coin, $C_i$, $10$ times and calculate the fraction of heads $v_1,v_2, .., v_{1000}$ for each coin. Now, each ...
0
votes
0answers
23 views

Finding a (tighter) sufficient condition on the standard deviation of a random variable

Let $\tilde{\phi}$ be a non-negative random variable with a mean normalized to $1$, with $F(\phi) := \Pr(\tilde{\phi} \leq \phi)$ denoting its CDF. $F(\phi)$ is assumed to be twice continuousy ...
0
votes
0answers
24 views

Bounding the standard deviation of a random variable

I have the following problem. Let $\tilde{\theta}$ be a non-negative random variable with twice continuously differentiable cdf $F(\theta) := \Pr(\tilde{\theta} \leq \theta)$ and $E(\tilde{\theta}) = ...
1
vote
1answer
125 views

How prove this distributions inequality $cov(\theta_{i},\theta_{j})\ge 0$?

Question: let random variable $\theta$ has dendity $f_{\phi}(\phi)$,and the random vector $\theta=(\theta_{1},\theta_{2},\cdots,\theta_{n})$,such $\theta_{i}|\phi$ are all independent from each ...
1
vote
1answer
37 views

Bounding the integral of the tails of a random variable.

I found an argument like this in a book, but I couldn't understand how we got this bound. Suppose $X_n$ is a sequence of random variables. For some $\delta > 0$ and all $n \geq 1$, $$ \int_{|X_n| ...
2
votes
1answer
69 views

Concentration inequality for the median

Most concentration inequalities talk about deviation of the sample mean from the population mean. Is there a result bounding the probability of deviation of the sample median from the median of the ...
1
vote
1answer
69 views

Tail probability of the $\chi^2$ distribution

Ho to prove that $$ \int_{2s\epsilon^{-2}}^{\infty}\frac{1}{\Gamma(d/2)2^{d/2}}x^{d/2-1}e^{-x/2}dx \leq const.\epsilon^{-d}\exp(-\epsilon^{-2}s) $$ holds for $\epsilon >0$ sufficiently small? Here ...
4
votes
0answers
94 views

A tight lower bound for the entropy of the XOR of two random variables

Let $U$ be the uniform random variable over $n$-bit binary strings, and let $X$ be another random variable that is dependent on $U$ and ranges over $n$-bit binary strings. Assuming $I(X;U) \le ...