# 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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### Inequality relating expected value and tail probability

I am currently working through Scornet2015 - Consistency of Random Forests. I'm having trouble understanding a specific inequality that is used in the proofs without further explanation. I am assuming ...
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### What kind of formal guarantee does a confidence interval provide after an observation?

According to my understanding, $C(X)$ is a random variable defined as follows: Let $\mathcal{P}$ be a family of distributions (defined by user). Let $\alpha>0$. Let $\theta$ be some parameter of a ...
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### Is it true that $\langle X^4\rangle \ge 3 \langle X^2\rangle^2$?

Consider a real random variable $X$ with zero mean. Does the following inequality hold in general? $$\langle X^4\rangle \ge 3 \langle X^2\rangle^2$$ I'm not sure how to prove this or if a counter-...
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### Number of samples for Hoeffding's Bound with Gaussian R.V

I am trying to obtain the required number of sample $n$ for a given confidence interval $\alpha$ and $X_1 ... X_n$ which are Gaussian rv with $\mu$ mean and $\sigma^2$ variance. I know that \begin{...
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### Probability of Roots outside the unit disc

Consider a random polynomial $p(z)=\sum_0^n A_i z^i.$ where $A_i,i=0,12,3,..,n$ are iid uniform variables in the interval $(0,1).$ I want to show that that the probability of the root with min modulus ...
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### Deriving and visualizing bounds on the conditional probability given marginal conditionals

Suppose I have a binary outcome $Y$ and two binary covariates $X_1$ and $X_2$ with distribution $P(X_1, X_2)$ which I know. In addition to $P(X_1, X_2)$, I know $P(Y\mid X_1)$ and $P(Y\mid X_2)$. I ...
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### probability of being within a 2D grid cell

Consider two joint-normal gaussian variables A and B with known probability density. A and B form a point in 2D space. What is the probability that the sample point falls within a rectangular grid ...
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### Holder's inequality in the case of $L_1$ and $L_{\infty}$ norm

I am referring to Wainwright's High-Dimensional Statistics book, where at some point it is deduced that \begin{equation} \frac{w'X\Delta}{n}\leq \left\lVert\frac{w'X}{n}\right\rVert_{\infty}\lVert\...
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### Proving Chebyshev's Inequality

I'm working on proving Chebyshev's Inequality. I watched this YouTube video and it almost makes sense. There is one step in the proof I don't understand. Using Markov's Inequality you substitute ...
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There are numerous issues that have been identified both in the theory and practice of $p$-values, including the arbitrariness of confidence levels, interpretation, and tail-risk in the ...