0
votes
1answer
25 views

Inequalities for the probability of union and intersection of events

Prove that $\min(1, P(A)+P(B))\ge P(A\cup B)$ $\min(P(A),P(B))\ge P(A\cap B)\ge \max(0,P(A)+P(B)-1)$ Where $A$ and $B$ are events. I don't know how to prove them; Can you give me a hand please?, ...
1
vote
0answers
17 views

Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
2
votes
0answers
119 views

Proving probability inequality (how to return to Chebychev?)

Supposing $X$ is a random variable, $X>0$, $E[X^2]<+\infty$, $\lambda \in (0,1)$, I have to prove the following inequality. $$P[X>\lambda E[X]] \geq (1-\lambda)^2 \frac{E[X]^2}{E[X^2]}$$ ...
0
votes
1answer
24 views

Applying Markov's inequality to a sequence of random variables

Does the Markov inequality also work for infinite $a$ or only for constant $a$? More precisely: If $X(n)$ is a sequence of random variables and $f(n)$ is some sequence of numbers,is it allowed to ...
0
votes
0answers
45 views

An expectation inequality [closed]

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ...
1
vote
2answers
44 views

Upper bound on the entropy of a sum two random variables

Let $X$ be a random variable such that $|X| \leq A$ almost surely, for some $A > 0$. Let $Z$ be independent of $X$ such that $Z \sim {\cal N}(0, N)$. My question is: How large can the entropy ...
0
votes
1answer
33 views

Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
2
votes
1answer
175 views

Entropy of the product of two random variables

Consider a random matrix $X$ and a random vector $Y$. Let the Shannon entropies $H(X) = H(Y) = n$. Is there a simple upper bound for entropy $H(XY)$? I believe $H(XY) \leq 2n$ as that is a simple ...
1
vote
1answer
32 views

How to show that $\Phi(1-x)^{-1} =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
2
votes
1answer
48 views

Expectation related to Normal distribution and its density

Given $\sigma^2>0$. Let $Z\sim N(0,1)$ and $\Phi$ be the cumulative standard normal with density function $\phi$. I wish to show that $$ E\left(\frac{Z^2}{[\phi(\sigma Z)]^2}\Phi(\sigma ...
0
votes
1answer
28 views

Limits for expected value in a proof

I have a small step in a proof, that I'm not sure if I got it right. We have given the function $f(s):=\mathbb{E}[e^{\lambda S (s-1)}]$ where $S$ is a random variable such that: ...
12
votes
2answers
132 views

Is there a probabilistic proof of the inequality $4p(1-p) \leq 1$ for a probability $p$?

Let $p\in(0,1)$. The inequality $4p(1-p)\leq 1$ is very easy and elementary, but I wonder if there is a probabilistic proof of it. By that, I mean constructing a “natural” probability space and an ...
0
votes
0answers
20 views

Vysochanskij Petunin vs. Cantelli inequality for random variables

The well known Cantelli inequality states: $$Pr(|X-\mu|\ge\alpha)\le\frac{2\sigma^2}{\sigma^2+\alpha^2}$$ where $X$ is a real valued random variable, $\mu$ the mean value and $\sigma^2$ the variance ...
0
votes
1answer
37 views

Approximation in Normal distribution random variable

Let ${X_n : n \geq 1}$ be independent $\mathcal{N}(0,1)$ random variables. How do we get the following approximation?
2
votes
1answer
43 views

Bessel's inequality for expected value

Let $X_1, X_2,\ldots$ be independent random variables with expected value $\mathbb{E}[X_i]=0$ and variance $V[X_i]=1$. Let $Y$ be another random variable, such that $\mathbb{E}[Y^2] < \infty$. I ...
-2
votes
1answer
43 views

Markov/Chebyshev's inequality Problems

Let $X$ and $Y$ be two random variables for which $ E(X)=75 $, $ E(Y)=75 $, $\mathrm{var}(X)=10$, $\mathrm{var}(Y)=12$, $\mathrm{cov}(X,Y)=-3$ (i) Find and upper bound to $P(|X-Y| \ge ...
0
votes
2answers
25 views

How is this algebraic step justified. (Inequalities)

I don't understand why this is allowed or the logic behind it: $P[X^2 - 2X < 8] = P[x^2 -2X + 1 < 9] = p[ (X - 1)^2 < 9 ] $ $P[-3 < (X - 1) < 3]$ (this step right here). What is the ...
0
votes
1answer
38 views

How to show $P[X\geq k] \leq (\frac{\lambda e}{k})^k e^{-\lambda}$ (X is Poisson random variable)

Let $X$ be a Poisson random variable with $\lambda > 0$ Show $\mathbb{P}[X\geq k] \leq (\frac{\lambda e}{k})^k e^{-\lambda}\qquad, \forall k \geq \lambda$ I'm having quite some trouble to show ...
0
votes
0answers
18 views

Proof of inequality between density functionals

I was wondering if there is an easy way to find sufficient conditions for the following inequality to hold $$ \int f(x,y)^2 \:\mathrm{d}x \:\mathrm{d}y - \int f(x)^2 f(y)^2 \:\mathrm{d}x\:\mathrm{d}y ...
0
votes
0answers
16 views

Finding conditions for joint probability density larger than the product of marginals

I was wondering if you could help me out. I have a joint probability distribution with density $f(x,y)$ and marginals $g(x)$ and $h(y)$ defined over the real line. Now, I would like to find a class of ...
1
vote
1answer
63 views

Convergence of Random Series

Let's say you have $X_1, X_2, ....$ independent real valued random variables and let $S_n = X_1 + ... + X_n$. Do you know how we can show that $P(\sup_{n\geq 1} |S_n| > 4\epsilon) \leq 4 ...
0
votes
0answers
52 views

Simple Chernoff bound

Im studying a paper where the following statement is done: The probability that a round is a success is at least $1-p^{-1}$ for $p\in\mathbb{N}^{\ge2}$. A simple Chernoff bound shows that the ...
0
votes
0answers
6 views

Is the following a proper application of Hoeffding's inequality?

I'm somewhat shaky on my probability and was wondering if someone could double-check that the following is a valid application of Hoeffding's inequality. Suppose that I have a random variable $X \sim ...
0
votes
0answers
40 views

Reference for theorem? Inequality of integrals of increasing function over two distributions

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$. This seems like it must be true: $$ ...
0
votes
1answer
37 views

is it possible to find X and Y such that E[X] is positive, Y is positive and E[XY] is strictly negative?

If $X$ is an integrable real random variables such that $E[X] \ge 0$ and $Y$ is a positive integrable random variable is it possible that E[XY]<0 ?
1
vote
0answers
39 views

One-Sided Bivariate Chebyshev Inequality

Let $X$ and $Y$ be random variables with finite means $\mu_X$ and $\mu_Y,$ finite variances $\sigma_X^2$ and $\sigma_Y^2,$ and correlation $\rho.$ Let $A$ be the event that $X \leq \mu_X + k\sigma_X$ ...
0
votes
1answer
37 views

Error Term of Chebyshev inequality?

Chebyshev inequality tells us that $$Pr[|X-E[X]|\geq a]\leq \frac{Var[X]^2}{a^2}$$ Do you know an Expression (or a paper where this Expression is mentioned) for the error term?
-4
votes
1answer
56 views

If $E(X)=0$, then $2E(|X|)\le\text{Var}(X)+1$ [closed]

If $E(X)=0$, $E\left(X^2\right)<\infty$, then $$2E(|X|)\le\text{Var}(X)+1.$$
2
votes
0answers
29 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
54 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
45 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
60 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
21 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
101 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
41 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
32 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
36 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
42 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
43 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
75 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
86 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
32 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
42 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 ...
1
vote
1answer
87 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?
3
votes
1answer
51 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
251 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
41 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
42 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
66 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
62 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, ...