# Tagged Questions

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### 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 ...
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### Show $P(h(X)\ge a)\ge\frac{E(h(X))-a}{b-a}$

Show that, if $h:\mathbb R\to[0,b]$ and $0\le a< b$ then, $\displaystyle P(h(X)\ge a)\ge\frac{E(h(X))-a}{b-a}$ So $h$ is nonnegative and bounded. If $a=0$ then the inequality holds. because ...
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### DKW-style $\ell_{\infty}$ bounds for sum of i.i.d. random functions: $\to [0,1]$

Let $\mathbf{G}$ be the set of (edit: convex) functions $g: X \to [0,1]$, where $X$ is a compact subset of $\mathbb{R}^d$ or something like that. Suppose I have a distribution $D$ on $\mathbf{G}$. ...
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### probability theory proof of exponential chebyshev inequality

This is a question about my homework. I am not sure about what is exponential Chebyshev inequality, also how do I get rid of the absolute value and prove it directly by PDF? As well as the ...
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### Generalized inverse of the cdf applied to a random variable equals the random variable itself almost surely?

first of all I apologize for the awful title but I really did not know how to formulate a precise question. Consider the following setup. Let $F$ be the distribution function of a random variable ...
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### Two Uniform Independent Random Variables: When is one greater?

You have two independent random variables: $X$ and $Y$, which are both uniformly distributed over $(0,1)$. Consider the inequality $X^2- 4Y < 0$. What percentage of the time is the inequality ...
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### 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 $X \leq \mu_X + k\sigma_X$ ...
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### 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 ...
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### 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 ...
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### Typo on Wikipedia's entry on Hoeffding's inequality?

In Wikipedia's entry on Hoeffding's inequality, they state that if $\overline{X} := \frac 1 n \sum_{i=1}^n X_i$, then $$P(\overline{X}-E[\overline{X}] \ge t) \le \exp (-2n^2 t^2)$$ if we assume for ...
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### 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 ...
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### Proof of Mrs. Gerber's Lemma Using Convexity and Jensen's Inequality

Can anyone give a proof of the Mrs. Gerber’s Lemma for the scalar case: $$\ H^{-1}(H(Y|U)) \ge H^{-1}(H(X|U))*p$$ where $$\ a*b = a(1-b) + b(1-a)$$ $$\ X\ ,\ Y$$ are binary random ...
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### 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 ...
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### Problem on inequality

Prove that, $E|X|^p < \infty$ iff $\sum_{k=1}^{\infty}k^{p-1}P\{|X| \geq k\} < \infty$. Where E is the expectation and P is the usual probability measure. There was a mistake one it's correct.
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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 $n \geq 1$, $$\int_{|X_n| ... 2answers 94 views ### Strange deduction about relation of median and mean On his blog T. Tao's proves the following concentration inequality, due to Talagrand. Let K>0, and let X_{1},..., X_{n} be iid complex random variables all bounded by K. Let ... 1answer 151 views ### Distance between mean and median I want to solve the following problem in T.Tao's random matrix theory book. Let X be a random variable with finite second momment. A median M(X) of X saisfies ... 1answer 86 views ### Application of Markov's inequality let h\colon \mathbb{R} \to [0,\alpha] be a nonnegative bounded function. Show that for 0\leq a<\alpha that the following holds: Pr(h(X)\leq a) \geq \frac{E[h(X)]-a}{\alpha-a} ... 1answer 105 views ### Lower bound of Fourier transform We know the Fourier transform of the Gauss-function: \displaystyle\int_{\xi\in\mathbb{R}^d}e^{-\pi\, C\,|\xi|^2}e^{2\pi i \xi\cdot X}d\xi=C^{-d/2}e^{-\, \pi\, |X|^2/2} for any C>0. Then ... 1answer 75 views ### Esséen concentration inequality I want to prove the following: Let X be a random variable taking values in \mathbb{R}^d. Then for any r>0, \epsilon >0: \displaystyle \sup_{x_{0}\in \mathbb{R}^d}{\bf{P}}(|X-x_{0}|\leq ... 1answer 107 views ### Lower bound on the probability of maximum of n i.i.d. chi-square random variables exceeding a value close to their number of degrees of freedom I am wondering if there is a tight lower bound on the probability of a maximum of n i.i.d. chi-square random variables, each with degree of freedom d exceeding a value close to d. Formally, I ... 1answer 258 views ### Average absolute value of sum with Rademacher random variables Let a_1, \ldots, a_n  be independent Rademacher random variables with distribution P(a_i=1) = P(a_i=-1) = \frac 12. Estimate from below$$E \left|\sum_{i=1}^n a_i\right|.$$I've reduced this ... 0answers 70 views ### Metric on the set of CDFs with finite p-th moment Let \mathcal{F}_p, p \ge 1, be the set of all cumulative distribution functions of real valued random variables whose p-th moment is finite. I'm looking for a metric on \mathcal{F}_p and ... 0answers 101 views ### Normal distribution inequality Let n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}, and N(x) = \int_{-\infty}^x n(t)dt. Prove the following inequality.$$(x^2+1)N + xn-(xN+n)^2>N^2$$where the dependency of n and N on ... 1answer 78 views ### How to prove this simple inequality? Please help me to prove this inequality. Suppose X and Y are independent and EX=EY=0, then we must have E(|X|) \leq E(|X+Y|). Thanks. 0answers 370 views ### Azuma's inequality to McDiarmid's inequality? I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ... 0answers 97 views ### Inequality of covariances between a bivariate normal vector and its indicator functions Why holds for a standardized bivariate normal vector Z:=(Z_1,Z_2) that $$|\operatorname{cov}(Z_1,Z_2)|\geqslant |\operatorname{cov}(1\{Z_1\leq u\},1\{Z_2\leq u\})|?$$ ... 2answers 77 views ###  E\left( \left|\frac{1}{n}\sum_{j=1}^n X_j\right|^p \right) \le \left( \frac{1}{n}\sum_{j=1}^n E(|X_j|^p)^{1/p} \right)^p The following is problem 14 of section 3.2 from Chung's "A Course in Probability Theory". If p>1, we have$$\left| \frac{1}{n}\sum_{j=1}^n X_j \right|^{p} \le \frac{1}{n}\sum_{j=1}^n |X_j|^p$$... 1answer 62 views ### P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2 for 0\le \lambda \le 1 If E(X^2)=1 and E(|X|)\ge a >0, then P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2 for 0\le \lambda \le 1. I can see from the well known inequality E(|X|) \le E(|X|^2)^{1/2} that it must be the ... 0answers 88 views ### A question about the stability of a property of the normal distribution Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution P on \mathbb{R} with the property that with X, Y ... 0answers 102 views ### a simpler proof for an inequality in probability I am trying to show that for any two real X,Y iid there holds that$$ P(|X - Y| \le 2) \leq 3P(|X - Y| \le 1)  I am getting nowhere with this and so I did some digging and found the following ...
How to show this inequality: If $\mathbb P (X \in [a,b]) = 1$, then $\operatorname{Var}(X) \leqslant \frac{(a-b)^2}{4}$. Thank you!
### Probability Theory: How to prove inequality $\mathbb{E}|X - m|^3 \leq \mathbb{E}|X|^3 (1 + \frac{m}{\sigma})^3$
Let's define $X$ - random variable with $F(x)$ distribution function. Also, denote $m = \mathbb{E}X$ and $\sigma^2 = \mathbb{D}X$. Suppose, that $m>0$. How to prove this inequality in these ...