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Mathematics

Questions tagged [variance]

For questions regarding the variance of a random variable in probability, as well as the variance of a list or data in statistics.

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1 vote
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Given a finite (multi)set of elements $\{x_1, \ldots, x_n\}$ the arithmetic mean $\mathsf{AM}$ is less than or equal to the maximum element call it $\max$. In otherwords, $\mathsf{AM} \leq \max$. But ...
3 votes
1 answer
50 views

Let $\varepsilon_1, \dots, \varepsilon_n$ be independent random variables with $E(\varepsilon_i) = 0$. Let $f: [0,1] \to \mathbb{R}$ be a Lipschitz function with constant $K > 0,ドル i.e., $$|f(x) - f(...
1 vote
0 answers
39 views

I understand that there is a similar question posted on the forum which talks about a unit circle and the solution can be extended to solve this question but I want to prove it by mathematically ...
2 votes
0 answers
59 views

I have a question when reading R. H. Shumway and D. S. Stoffer's Time Series Analysis and Its Application With R Examples, 5th edition. On page 181, section 4.1, it's said that Note that, if in (4.4),...
2 votes
1 answer
62 views

Let $x_1 \leq x_2 \leq \cdots \leq x_n$. Define $$ S_n^2 = \frac{x_1^2+x_2^2+\cdots+x_{n}^2}{n}-\frac{\left(x_1+x_2+\cdots+x_{n}\right)^2}{n^2}, $$ the variance, and $$ S_{n-2}^2=\frac{x_2^2+x_3^2+\...
2 votes
0 answers
37 views

Assume that $X,Y,Z$ are independently normally distributed (with potentially different mean and variance). Are there some "nice" formulae for \begin{align*} & \mathrm{Cov}\left(\mathrm{...
0 votes
0 answers
48 views

The question is taken from Achim Klenke's Probability Theory: A Comprehensive Course Section 5.1. There Blackwell-Girshick's equation is stated and proved with the assumption of independence of the ...
0 votes
2 answers
105 views

When working with a set of elements, one may wish to identify a subset whose variance does not exceed a given threshold. One possible approach is to examine smaller subsets first; if these subsets ...
0 votes
3 answers
109 views

I looked at the rules, I think it's not wrong to ask people to just explain something to you. (I hope the post won't get flagged). My knowledge level is as much as high-schooler. But I've been ...
1 vote
1 answer
102 views

Consequences of Isserlis's theorem: Are these formulas right? $E[e^{-iX}],ドル $E[\cos(X)]$ and $E[\sin(X)]$ Recently I made this question about the following consequence of Isserlis's theorem: $$E[e^{-...
1 vote
0 answers
60 views

Consider an $\large n \times n$ order matrix $\large M$. The $\large i,j$-th entries of the matrix $\large M,ドル let's say, $\large X_{i,j}$ is an i.i.d random variable ($\large \forall i,j$) following ...
0 votes
0 answers
108 views

The question is as follows: A box contains 2 red and 3 blue balls. Two balls are drawn successively without replacement. If getting ‘a red ball on first draw and a blue ball on second draw’ is ...
1 vote
1 answer
60 views

I'm trying to understand the diffusion models defined in continuous time as in this paper (https://arxiv.org/pdf/2107.00630). What I'm struggling is inducing the variance part of $$q(\mathbf{z}_t|\...
1 vote
0 answers
41 views

Let $X$ be a positive valued random variable, say with law $\mu$ on $[0,\infty),$ and for any $\lambda>0,$ let $\mu_{\lambda}(dx)$ denote its Cramer's transform, i.e.$$\mu_{\lambda}(dx):=\frac{e^{-\...
0 votes
0 answers
36 views

I read about, or saw in a video, a method to reduce variance in Monte Carlo integration which was particularly strong in moderate dimensional integrals, but I can't remember the name. The only ...

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