1
vote
1answer
31 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
44 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
26 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
127 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
16 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
36 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
40 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
23 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
35 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
50 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
38 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
33 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
53 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
44 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
59 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
87 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
37 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
42 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
67 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
83 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
30 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 ...
1
vote
1answer
79 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
40 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
222 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
41 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
65 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
59 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
52 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
100 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
125 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
66 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
61 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
3answers
140 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
42 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.