# 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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### Bounding the Variance of a Product of Dependent Random Variables

I want to bound from above the variance of the following function: $$g(w) = \|\sigma'(\langle z, w \rangle )w\|^2$$ where $w \in R^d$ is a vector of $d$ i.i.d random variables $w_i$ of normal ...
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### Stick breaking calculate cdf of breaking point [closed]

We have a stick of 1m that geats split at a random point whereas the breaking point is Uniformly distributed between $[0.1]$ . Whats the cdf of $X$ where $X$ gives us the breaking point of the stick + ...
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### Some questions about variance and expectation values

I'm reading Probability and Statistics with R by Ugarte, Militino, and Arnholt and there's a few questions I have related to variance. They state: $Var[X] = \sigma^2_X=E[(X-\mu)^2]=E[X^2]-\mu^2$ ...
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### Estimator and Variance of powers of a random variable [closed]

$X$ is a random variable, $n$ is an integer, let $\mu_n = E(X^n)$. How could we construct: Estimator $\hat{\mu}_n$? $\operatorname{Var}(\hat{\mu}_n)$ Appceciated for your kind instruction.
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### Variance of Sum to Sum of Variance

There are some proofs of this idea elsewhere but I am trying to demonstrate it to myself with a specific example so as to understand the derivation of the formula for the Standard Error of sample ...
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### What is $\operatorname{Var}(f_1 \mid f_2 > 0)$ when $(f_1,f_2)$ is multivariate Gaussian?

I know $(f_1,f_2)$ follows a multivariate Gaussian distribution, with mean $m$ and covariance matrix $V$. I would like to compute $\operatorname{Var}(f_1 \mid f_2 > 0)$. There is a formula to ...
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### Variance and Expectation of Little o

Let $X$ be a random variable dependend on some parameter $t$. with finite expectation and variance for all $t$. How can I compute $E[X(t) + o(t)]$ and $V[X(t) + o(t)]$ for $\rightarrow 0$? From ...
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### Asymptotic moments of the first-passage time

I am reading a paper (Predictability of escape for a stochastic saddle-node bifurcation: When rare events are typical, Herbert, Bouchet, 2017 Physical Review E), which is related to Kramers escape ...
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### The expectation and variance of the $\infty$-norm $\|\hat{\mu}-\mu\|_{\infty}$. [closed]

Suppose the i.i.d. sample $\{X_i\}_{i=1}^n$ follows a $p$-dimensional normal distribution $N_p(\mu, \Sigma)$, where $\Sigma>0$ and the eigenvalues of $\Sigma$ is bounded away from $0$ and $\infty$. ...
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### Is there a way to find out which option was mostly answered by a student in a quiz? [closed]

Consider a quiz with 5 questions each question having 4 choices ( A, B, C, D). Is there a way to mathematically find which option was answered the most by a student if we can assign some values to ...
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### Confidence interval of the population mean when variance is known and the sample size is large enough

We observe the height of the population of a city as identically distributed random variables and Variance $\sigma^2=7.3$ cm and the mean of the height of 200 randomly chosen inhabitants is 176.2 cm. ...
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### Double differentiation of characteristic function of Normal random variable

Knowing that a one-dimensional random variable $\Gamma$ is Gaussian if it has the characteristic function $$\mathbb{E}\hspace{0.15cm}e^{i\xi\Gamma}=e^{im\xi-\frac{1}{2}\sigma^2\xi^2}\tag{1}$$ for some ...
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### How do you prove the mean of variance of the return between 2 stocks?

I've studying the "Black-Scholes-Model" from a book. I'm letting S(x) be the price of the stock at time x, S(y) be the price of the stock at time y, r be the stock's expected rate of return ...
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### Linear formula for measuring spread of data

TLDR: Is there any formula, that measures the spread of a data set (like variance or std deviation), but is linear? Longer version: I'm working on an optimisation problem and the objective of that ...
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### Variance of Univariate Gaussian Mixture

Let $\mathbb{P}_1,\dots,\mathbb{P}_n$ be univariate Gaussian measures with respective means $m_1,\dots,m_n \in \mathbb{R}$ and respective variances $\sigma_1,\dots,\sigma_n$. Let $r_1,\dots,r_n$ be ...
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### Determine all $\overrightarrow{a}$ for an unbiased estimator of the variance

consider a random variable $X$ and stochastically independent repetitions $X_1,...,X_n$ of $X$. For each vector $\overrightarrow{a}=(a_1,...,a_n) \in \mathbb{R}^{n} \text{ with } a_i > 0$ we ...
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### Finding the Variance

A vat contains millions of marbles. $20\%$ of them have the number “$6$” carved on them, and the rest have the number “$17$” carved on them. A man possesses $3$ eggs. A marble is drawn from the vat, ...
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### Rough rule of thumb for when to use $N$ and when to use $N-1$ when calculating variance.

I don't really understand the whole concept of estimators and biased/unbiased etc with regards to calculating the standard deviation. Is there a rough rule of thumb that I can use to know when I ...
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### Why don't we just take the root of the numerator instead of taking the root of the whole thing in the expression for standard deviation?

The equation of the standard deviation of a dataset is given by $\sqrt{\frac{\sum{(x_{i} - \bar{x}})^2}{N}}$. Why is that the case and why can't we use $\frac{\sqrt{\sum{(x_{i} - \bar{x}})^2}}{N}$ ...
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### Chebyshev inequality but with normalized maximum estimation error

Consider random samples $X_1,X_2,...$ being identical and independent copies of $X$. Their sum $S_n=\Sigma ^n_{i=1}X_i$. The mean of $X$ is $\mu$ and variance is $\sigma^2\lt\infty$. Regarding the ...
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### What kind of deviation is the $sx$ button on my calculator?

I'm learning about standard deviation and I notice there is another kind of deviation option on my calculator. The standard deviation is denoted by $\sigma x$ on my calculator while the other kind of ...
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### Why is variance defined as $\sum\limits_{n} |\mu -x_i|^2$ and not $\sum\limits_{n} |\mu -x_i|$?

If we wanted to measure how much the values $x_1, \ldots ,x_n$ of a sample differ from the mean $\mu$, it seems more intuitive to me to use the formula $$\frac{\sum\limits_{n} |\mu -x_i|}{n}$$ ...
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### Finding the conditional variance of two discrete random variables in a coin flip

I'm having trouble calculating the conditional variance for a problem: Suppose we flip a fair coin 5 times in a row. We denote the random variable $X$ to be the total number of heads in the end, and ...
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### When is $\sqrt{k}\operatorname{Var}\left(\sum_{i=1}^{k}X_{i}\right) \leq \sum_{i=1}^{k}\operatorname{Var}(X_{i})$ true?

Assume we have $k$ dependent random variables $X_{1}, \dots, X_{k}$ with $\operatorname{Var}(X_{i}) < \infty$. In which case \sqrt{k}\operatorname{Var}\left(\sum_{i=1}^{k}X_{i}\right) \leq \...
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### Variance of $X$, an uniformly random sum from a finite set $S$. [closed]

This is from my class. Can I have an explanation of what going on in the last equality (i.e. $\operatorname{Var}\left(\varepsilon_{i}\right) s_{i}^{2}=\frac{1}{4} \sum_{i=1}^{k} s_{i}^{2}$)? ...
### What is the variance of $\frac{1}{n-1}\sum_{i=1}^{n-1}(x_{i+1}-x_i)^2$?
Consider a length $n$ vector $\mathbf{x}$ containing $n$ i.i.d. observations $\{x_i\}_{i=1}^n$ of a random variable $X$ with zero mean and unit variance. Let $\mathbf{z}$ be a length $n-1$ vector ...
So I've been given the following exercise: Let $X$ and $Y$ be two random variables. Let $Var(X) = 2$ and $Cov(X, Y) = 1$. Compute $Cov(5X, 2X + 3Y)$. How do I do this when I don't have the second ...