# Questions tagged [probability-inequalities]

Probability Inequalities are useful for bounding quantities that might otherwise be hard to compute. A related concept is a concentration inequality, which specifically provides bounds on how far a random variable deviates from some value.

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### Error in applying Chebyshev's inequality

I'm trying to solve a problem using Chebychev^' s Inequality: "Suppose that X is a random variable with mean and variance both equal to 20. What can be said about P(0<X<40)?" P(|X-μ|≥...
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### Concrete bound of expected value of a difference of I.I.D. Uniform Random Variables

In the following, $X,X_1,X_2,\dots X_n$ are I.I.D. uniform random variables in $[0,1]^d$ in $\mathbb{R}^d$. The problem I am attempting to solve is Exercise 2.4 from Gyorfi's "A distribution free ...
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### How to derive Paley-Zigmund Inequality proof

The Paley-Zygmund inequality is given by \begin{equation} \operatorname{P}( Z > \theta\operatorname{E}[Z] ) \ge (1-\theta)^2 \frac{\operatorname{E}[Z]^2}{\operatorname{E}[Z^2]} \end{equation} I ...
1 vote
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### Finding a consistent sequence of estimators such that $\lim_{n\to\infty} E_\theta[(W_n-\theta)^2]\ne 0$

There are many ways to check if a sequence of estimators is consistent. By definition, a sequence of estimators $W_n = W_n(X_1,X_2,\ldots,X_n)$ is a consistent sequence of estimators of the parameter ...
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### 0-1 laws in random graphs: probability $\beta$ is large if $k$ is large

How has the author derived here on the page 3 in the context of random graphs and 0-1 laws that $\beta$ is large if $$k\geq ((\frac{2}{\alpha})\log n)^{\frac{1}{2}}$$ ? What I did is this: I've ...
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### monotonicity of sample averages tails as a function of sample size

Let $X_1,...$ be iid mean zero random variables. The LLN says $\overline{X}_n\to 0$. I am curious if the following is true: is $P(|\overline{X}_n|>x)>P(|\overline{X}_{n+1}|>x)$ for $x$ ...
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### Show $(E|X|^2)/(E|X^2|) \leq P(X \not =0)$

I'm looking to show this inequality is true, and in turn use it to conclude the second moment method's bound. Show that $\frac{E|X|^2}{E|X^2|} \leq P(X \not =0)$. Again, I'm not supposed to use second ...
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### Does maximizing Jensen–Shannon divergence maximize Kullback–Leibler divergence?

Does maximizing the Jensen–Shannon divergence $D_{\mathrm{JS}}(P \parallel Q)$ maximize the Kullback–Leibler divergence $D_{\mathrm{KL}}(P \parallel Q)$? If so, I'd like to be able to show that it ...
1 vote
Let $X$ be a random variable with support $(0,\infty)$. All I know about $X$ is the support, finite higher moments, and $\mathbb{E}(X)=\mu$. I am trying to come up with a more tractable upper bound ...
### Prove that $E[\log(\alpha X_t^2)] < 0$ implies $\alpha < 3.5622$ with $X_t \sim N(0,1)$
I am trying to prove this statement: If $X_t \sim N(0,1)$ then $$E[\log(\alpha X_t^2)] < 0 \implies \alpha < 3.5622$$ which is a a necessary condition often found in textbooks for strict ...