# 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. Gerberâ€™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 ...
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.