# 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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### Why do we need Jensen inequality for variational autoencoders?

Just to clarify, I think I understand all the derivations in context of VAEs pretty well; however, there is one last thing that I need explained. There are multiple related derivations of the evidence ...
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### Solving for the parameter of an exponential distribution

Suppose I have a random variable $X$ where $X$ follows an exponential distribution of the following form: $$f_X(x) = \frac{1}{\lambda}e^{-\frac{x}{\lambda}}$$. I want to find the value of $\lambda$ ...
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### Product of a series and asequence of random variables

Suppose $\{a_n\}_{n \in N}$ be a bounded sequence positive numberse bounded by $b$ and $Z_j,j=1,2,3,...$ be a sequnce of complex random variables such that $E(|z_j|)<\frac{1}{\sqrt{(j+1)m}}$ for ...
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### For $Y \geq 0$, prove that $Pr(Y \geq k) \leq E(Y)/k$

Let $Y$ be a non-negative random variable, $k$ be any positive constant, show that $Pr(Y \geq k) \leq E(Y)/k$. My attempt (using integration by parts): \begin{align} \int_0^k y \,dF(y) &\leq E(Y) \...
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### How to prove this inequality?

For any nonnegative random variable $X$ independent of $U$ where $U \sim \operatorname{Uniform}(-t,t)$ and any $t\ge 0,$ $$P(X+U\ge t)\le\frac{E(X)}{2t}.$$ Any hints to prove this inequality?
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### Probability Conditioned on Inequality

Assume that $A \sim \mathcal{N}(0, 1)$, $B \sim \mathcal{N}(0, 1)$. I am trying to calculate $P(A \,|\, A < B)$. For the sake of this problem, we can assume that $A \perp B$, but (for obvious ...
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### Is my way of deriving a statistical test from Hoeffding's inequality correct?

I'm trying to deduce from samples of observations from two finite sets of random variables $X_{1}, ..., X_{n}$ and $Y_{1}, ..., Y_{m}$ that the expected values of the average of those random variables ...
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### Variance inequality for nested sets

Let $X, Y,$ and $Z$ are three random variables/vectors, and let $f(., ., .)$ is a real-valued, deterministic function. If $Z$ is independent of $\{X, Y\}$ (e.g., $X, Y, Z$ are independent) then \begin{...
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### Test for mean 0 based on Bennett inequality?

Bennett's inequality provides a bound on the probability for the sum of bounded mean 0 random variables to exceed a specified value. It makes no assumptions about distribution, and is stronger than ...
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### Markov Inequality for Sum

I don't have proper knowledge of probability, I was surfing around internet about Markov's inequality, I found a paper on JSTOR titled "The Markov Inequality for Sums of Independent Random ...
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### Bounding sum of quartic deviations from sample mean

[Cross-posted here with no answers for a few days] I came - to the very best of my knowledge from reading the source - across the following statement in The Jackknife and Bootstrap, Shao and Tu, p. 87:...
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### When is Jensen's Inequality strict?

For a homework problem, I have to prove that for a random sample $X_1, \ldots, X_n$, drawn from a population with finite variance $\sigma^2$, with sample mean $\bar{x}$ and sample variance $s^2$, that ...
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### Concentration inequalities for estimated least squares regression coefficients?

I would like to know what is the best concentration inequality we can use for the estimated least squares regression coefficients. Let $\hat \beta_0, \hat \beta_1$ be the estimated regression ...
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### Sum of continuous i.i.d random variables

Let $X_{1}, X_{2}, \ldots X_{N}$ be non-negative continuous i.i.d random variables such that the probability density function of each $X_{i}$ is given as f(x) = N e^{-Nx}. \end{...
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Can I break down $P(h \geq (A + B)$, given all $A,B,h$ are all random variables. Will the following rule works? $$P[h \geq (A + B)] = P(h\geq A) + P(h\geq B)$$ Actually, in one of my mathematical ...