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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24 views

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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1answer
21 views

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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26 views

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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18 views

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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19 views

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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29 views

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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17 views

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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29 views

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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68 views

Computing One-Pass Covariance with Respect to Another Covariance

According to the Wikipedia page on computing the One-Pass Covariance, for $X_{i,n}=(x_i, x_{i+1},..., x_{i+n-1})$ and $Y_{i,n}=(y_i, y_{i+1},..., y_{i+n-1})$, the covariance is: $ \begin{align} ...
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1answer
30 views

How to link the variance of the distance between two vectors to the variance of their norms?

I'm looking at a simplified case of this question where I have a random vector variable $X$ in dimension $k$. I know this about $X$: its mean $E[X]$, its covariance matrix, the mean of its $L^2$ norm $...
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33 views

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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2answers
43 views

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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1answer
31 views

What significance does the fact that mean minimises variance have?

Consider the variance in statistics, defined as $$\sigma^2 = \frac{\sum(x_i-m)^2}{n}$$ (assuming $f_i=1$ for all $i$). Let's differentiate this quantity. We obtain $$\frac{d \sigma^2}{dm} = \frac{-2 \...
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3answers
42 views

Variance of a discrete random variables that takes on 2 values.

Suppose I have a random variable that takes on a value of 10 with p(x=10)=.7 and a value of 20 with p(x=20) = .3. The E(X) = .7(10)+.3(20) = 13. The variance would be the expected value of the ...
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2answers
74 views

Given 3 heads or tails is a successful trial, what is the variance of number of tails until the first success?

We are flipping three fair coins simultaneously. Assume that each trial is independent. We define any trial as a success if three coins flipped are on the same side. Let a Random Variable M be the ...
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1answer
24 views

calculate expected value and variance of $Y := 1 − X$

I'm struggling with the following exercise: Given the random variable $X$ with expectation value $\mu$ and variance $\sigma^2$: What is the expectation value and variance of $Y := 1 − X$ Isn't it just ...
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1answer
45 views

Analytic expression for continuous-variable mutual information of uniform distributions

I want to quantify how mutual information depends on the variance of one of the variables. Here's a simple test I came up with $$X \sim U(0, 1)$$ $$Y \sim U(0, 1)$$ $$Z = (X + Y) / 2$$ where $U$ ...
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1answer
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Determining the convergence and correlation coefficient between two random variables which are sums of other random variables

Random variables $X_1,\:...,\:X_n$ are independent and have the same variance $\sigma ^2$. Let $U=3X_1+X_2+...+X_n$ and $V=X_1+X_2+...+X_{n-1}+2X_n$. Determine the correlation coefficient between U ...
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1answer
24 views

Covariance matrix in multivariate standard normal density

I am looking at the derivation of $f_{\vec{Y}}(\vec{y})$ where $\vec{Y}=A \vec{X}$ and $\vec{X}$ is a vector of i.i.d standard normal random variables. $A$ is an $n \times n$ non-singular matrix. ...
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2answers
25 views

Inferring Covariance / Correlation

Is it possible to derive Cov (Y,Z) from Cov (X,Y), Cov(X,Z) and the corresponding Variances? All 3 variables are assumed to be normal with 0 mean.
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1answer
32 views

bounds for conditional expectation and variance

let X and Y be random variables with density: $$f_{x,y}(x,y)=\frac{1}{\pi}\mathbf{1}_{\{x^2+y^2\le1\}}$$ Find $$E[X|Y] \& Var(X|Y)$$ So what I did was: $$E[X|Y]=\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}}\...
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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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13 views

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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1answer
14 views

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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1answer
10 views

expected value of vector

$X = [X_1X_2]^T$ $M = [ 1 $ $2]$$^T$ $∑$ = $[3$ $1,$ $1$ $4]$ $∑_{11} = 3 $ $∑_{12} = 1 $ $∑_{21} = 1 $ $∑_{22} = 4 $ I couldn't write as a matrix $E[X^TX]$ = ? I found $E[X_1^2+X_2^2]$ ...
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13 views

Finding percentage value of standard deviation when given number of elements in sample, and population's standard deviation

I'm faced with an exercise that goes as such: "A sample of 81 elements is taken from a population whose standard deviation is 3. What is the approximate value of the mean's sample deviation, in ...
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1answer
25 views

distribution of fractional Brownian motion increments

For a standard Brownian motion $(B_t)_{t \geq 0}$, we have $$B_t \sim\mathcal{N}(0, t),\quad B_t-B_s\sim\mathcal{N}(0, t-s)$$ Then, what are the distributions of increments for a fractional Brownian ...
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1answer
26 views

Why the variance of uniformly distribution is like that?

According to several reference, The variance of uniformly distribution is like below: $$\frac{1}{12} (b-a)^2$$ However, after calculating the variance from scratch: $$\sum _{x=a}^b \frac{\left(x-\frac{...
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25 views

Expected Values, Density and Variance of Trigonometric Functions

I was reading about Variance in Probability Theory textbook and then I was wondering about one thing: If we choose a random angle, let´s call it $\theta$, which would be uniformly distributed on $(0, \...
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44 views

Cramér–Rao bound

Let $\xi$ - random variable with probability density function is $$p(x;\theta):=\left\{\begin{array}{ll}e^{\theta-x},&x\geq\theta \\ 0,&x<\theta\end{array}\right.,$$ $\tilde\theta=\tilde\...
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1answer
31 views

How to measure variance of distances from origin

I'm trying to measure the sample variance of some data. Such data are 2D euclidean distances from the origin (0,0). Supposing to have the 2 components X and Y used to calculate the distance, it's ...
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28 views

Confidence intervals for ratios of variances of different populations and other confusing stats

I'm trying to solve an old statistics and probabilities exam exercise for my uni. I don't have a lot of time to learn the subject so I'm trying to figure out how to solve exercises from older tests. ...
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1answer
47 views

Expected number of red balls in urn of two-coloured balls with color substitution.

There are $m$ black and $n$ red balls in an urn. One randomly picks a ball from the urn. If the picked ball is the black one, person changes it with the red ball and returns to the urn. If the picked ...
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27 views

Variance of $X/\hat{\sigma}_X$

Imagine that we have $X_1, ..., X_n$ that are i.i.d. and we perform a standardization procedure for each variable, i.e.: Step 1. Subtract the mean: $Z_i' = X_i - \hat\mu$ Step 2. Divide by std: $Z_i =...
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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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59 views

Chebyshev inequality but with normalized maximum estimation error (cont.)

This is the continuation from : 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=...
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50 views

Delta method for asymptotic variance

Let $X_i$ be i.i.d. r.v. with $\sim Exp(\lambda)$ and for n to $\infty$ $$\sqrt{n} \cdot (Z_n - \theta) \overset{d}{\to} N(0,\sigma^2) $$ $\tilde{\lambda} = - ln(\bar{Y_n})$, where $Y_i = 1 \{X_i = 0\...
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43 views

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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28 views

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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2answers
61 views

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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1answer
123 views

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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0answers
22 views

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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1answer
72 views

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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19 views

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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1answer
47 views

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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1answer
25 views

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. $\operatorname{Var}\left(\varepsilon_{i}\right) s_{i}^{2}=\frac{1}{4} \sum_{i=1}^{k} s_{i}^{2}$)? ...
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27 views

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 ...
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0answers
27 views

Calculate covariance of 2 random variables when only given variance and cov

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 ...

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