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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.

34
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3answers
111k views

Determining variance from sum of two random correlated variables

I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated?
13
votes
0answers
239 views

Trimmed mean: Take $n$ i.i.d. Gaussians and remove largest $m$ and smallest $m$ points. What is the variance of the mean of the remaining points?

Let $n,m\in\mathbb{Z}$ with $0 \le 2m < n$. Let $X_1, \cdots, X_n$ be i.i.d. standard Gaussians and let $X_{(1)} \le X_{(2)} \le \cdots \le X_{(n)}$ denote their order statistics (i.e., $\{X_1, X_2,...
8
votes
4answers
166 views

Why does $ \operatorname{Var}(X) = E[X^2] - (E[X])^2 $

$ \operatorname{Var}(X) = E[X^2] - (E[X])^2 $ I have seen and understand (mathematically) the proof for this. What I want to understand is: intuitively, why is this true? What does this formula tell ...
8
votes
3answers
136 views

Intuition behind binomial variance

Suppose that I perform a stochastic task $n$ times (like tossing a coin) and that $p$ is the probability that one of the possible outcomes. If $K$ is the stochastic variable that measures how many ...
7
votes
2answers
12k views

What is the Difference between Variance and MSE?

I know that the variance measures the dispersion of an estimator around its mean i.e. $\sigma^2 = E[(X - \mu)^2]$ (the second central moment about the mean). But I'm not getting the meaning of the ...
7
votes
1answer
23k views

Proving $\operatorname{Var}(X) = E[X^2] - (E[X])^2$

I want to understand something about the derivation of $\text{Var}(X) = E[X^2] - (E[X])^2$ Variance is defined as the expected squared difference between a random variable and the mean (expected ...
6
votes
4answers
900 views

Variance of sine and cosine of a random variable

Suppose $X$ is a random variable drawn from a normal distribution with mean $E$ and variance $V$. How could I calculate variance of $\sin(X)$ and $\cos(X)$? (I thought the question was simple and ...
6
votes
2answers
7k views

Difference between Variance and 2nd moment

I understand that $Var(X) = E(X^2) - E(X)^2 $ And that the second moment, variance, is $E(X^2)$ How is variance simultaneously $E(X^2)$ and $E(X^2) - E(X)^2$?
6
votes
3answers
444 views

Is the usage of unbiased estimator appropriate?

Sometimes I find the usage of unbiased estimator quite confusing. For example, the unbiased estimator of variance:$$S^2=\frac{\sum (X_i-\bar{X})^2}{n-1}\,.$$ True, it is the expectation of variance. ...
6
votes
1answer
4k views

Expectation and variance of the Pareto distribution

Given the distribution funciton of the r.v. $X$ for $\alpha, \beta >0$ $$ F(x) = 1-\Big( \frac{\beta}{\beta +x}\Big)^{\alpha} $$ for $x \geq 0$ and $0$ elsewhere What is the ...
6
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2answers
220 views

F test vs T test. What is the real diference?

As I've learnt, a T - test is used to compare 2 populations’ means, whereas an F-test (ANOVA) is used to compare 2/> populations’ variances. At the end is this doing the same thing? My background is ...
6
votes
2answers
156 views

Axiomatic approach to the definition of variance

I'm trying to grasp the intuition behind the definition of variance. It seems plausible that we want to measure how much a random variable deviates from it's expected value. But why using the square ...
5
votes
3answers
2k views

Negative Variance

I have two independent variables $X$ and $Y$. $W=X-Y$ when $X\sim \mbox{Bernoulli}\left(1/2\right)$ and $Y\sim N(0,1)$. This puts $\operatorname{Var}(x)=1/4$ and $\operatorname{Var}(Y)=1$, but I have ...
5
votes
1answer
7k views

what is the variance of a constant matrix times a random vector?

$\newcommand{\Var}{\operatorname{Var}}$In this video is claimed that if the equation of errors in OLS is given by: $$u=y - X\beta$$ Then in the presence of heteroscedasticity the variance of $u$, will ...
5
votes
2answers
494 views

Formula for a “Fairness Variance”

Short question: Propose a formula to apply on x0, x1, x2, ..., xn that returns a number which can sort these 7 datasets in this order: Medium question: Given 3 ...
5
votes
2answers
92 views

Finding $\operatorname{E}(X)$ given that $\operatorname{E}(X) = \operatorname{var}(X)$, $\operatorname{E}(Y) = \operatorname{var}(Y)$, and $Y=3X+3$

$\newcommand{\E}{\operatorname{E}}\newcommand{\v}{\operatorname{var}}$Suppose a random variable $X$ is such that its expected value is equal to its variance. If $Y= 3X+ 3$ is also a random variable ...
5
votes
1answer
64 views

Expectation and variance of iterated dice rolling

For the rest of the post, we'll assume when I say "die" or "dice" that I mean a standard 6-sided die or some number of standard 6-sided dice. Suppose I roll a die and get a number $n$, and then I roll ...
5
votes
0answers
129 views

Supermartingale variance bound?

Suppose I have a supermartingale $$ \mathbb{E}[X_{n+1} \mid X_n, \dots , X_2, X_1] \leq X_n $$ There are 2 other conditions: bounded difference: $|X_{n+1} - X_n| \leq A$ with probability $1$ (...
4
votes
2answers
5k views

Why square a constant when determining variance of a random variable?

If I want to calculate the sample variance such as below: Which becomes: $\left(\frac{1}{n}\right)^2 \cdot n(\sigma^2)= \frac{\sigma^2}{n} $... My question is WHY does it become $$\left(\frac{1}{n}\...
4
votes
1answer
361 views

The data set $\{x_1,\dots,x_{10}\}$ has a mean $\mu=10$ and a standard deviation $\sigma=3$. Find the value of $\sum_{i=1}^{10}[(x_i-12)^2]$.

Problem The data set $\{x_1,\dots,x_{10}\}$ has a mean $\mu=10$ and a standard deviation $\sigma=3$. Find the value of $$\sum_{i=1}^{10}\left[\left(x_i-12\right)^2\right]$$ My solution Using ...
4
votes
2answers
125 views

If we have infinite variance, can the expectation 'mean' anything?

If we have a random variable $X$ with infinite variance $Var(X)$ then how can the expectation $E(X)$ be useful to us? In the sense that our values vary so much from it that it holds no relevance. ...
4
votes
3answers
232 views

Variance of couples seated across from each other at a rectangular table.

Let N couples be randomly seated at a rectangular table, men on one side and women on the other. Let X be the random variable describing the number of couples that end up being seated across from each ...
4
votes
2answers
305 views

Variance of X vs Variance of a binary function of X

Let $X$ be a random variable in $[0, 1]$ and $m$ its median such that $P(X \le m) = P(X \ge m)$. Define $\beta(X)$ as $$\beta (x) = \left\{ \begin{array}{c} \begin{align*} 1&,\space X \ge m; \\...
4
votes
2answers
723 views

If $\operatorname{var}(X) =0$ then $p( X = E (X)) = 1.$

Let $X$ be a random variable. Then $\operatorname{var}(X) =0$ implies $p( X = E (X)) = 1.$ Actually i need both directions. But i was able to show that $p( X = E (X)) = 1$ implies $\operatorname{var}...
4
votes
2answers
143 views

A simple variance question

Say I am paid 1 million dollars per pip on a six sided dice and I roll the dice one time. Now compare this to a game in which I get 1 dollar per pip on a dice, but I roll the die one million times. It ...
4
votes
3answers
374 views

Proof of Variance of the Irreducible Error

In Introduction to Statistical Learning, given the general form of a quantitative response between a set of predictor variables and a target variable $$Y=f(X)+\epsilon$$ and the general form for a ...
4
votes
3answers
83 views

A measure similar to variance that's always between 0 and 1?

Consider the following histogram, obtained from around 1000 measures of distance. As you can observe, most of the data appears near the mean arond the value 5-10. I also have some isolated samples ...
4
votes
2answers
110 views

Why square the term $X - \mu$ in the definiton of the variance?

Why is is the variance $\operatorname{Var}(X)$ of a random variable $X$ definied to be $\operatorname{E}[(X-\mu)^2]$? My professor said that you want the variance to be positive, but why not go for $...
4
votes
3answers
539 views

The relationship between sample variance and proportion variance?

I'm trying to see the relationship between the sample variance equation $\sum(X_i- \bar X)^2/(n-1)$ and the variance estimate, $\bar X(1-\bar X),$ in case of binary samples. I wonder if the ...
4
votes
1answer
40 views

Random variables and co-variance, Statistics 318

For the given example in the book John E. Freund's Mathematical Statistics with Applications, 8th edition, by Miller and Miller. ISBN: 9780321807090 I've highlighted using colors what numbers ...
4
votes
1answer
831 views

Simple Random Walk, proving variance of the walk at a stopping time

So the problem is that $S_N$ is a symmetric random walk and $T$ is a bounded stopping time. I have to show that the variance $$\text{Var}\left(S_T\right) = \mathbb{E}\left(S_T^2\right) = \mathbb{E}(T)$...
4
votes
1answer
192 views

Variance of the Euclidean norm under finite moment assumptions

Let $X = (X_1,X_2 \cdots X_n)$ be random vector in $R^n$ with independent coordinate $X_i$ that satisfy $E[X_i^2]=1$ and $E[X_i^4] \leq K^4$. Then show that $$\operatorname{Var}(\| X\|_2) \leq CK^4$$...
4
votes
1answer
81 views

variation of random variable greater or equal than sum sum variations of two independent conditional expectations

I found the following problem, which seems very simple, but I'm stuck concerning the ideas I can use. The statement is as follows: Let $X$ be a random variable with finite expectation and let $\...
4
votes
1answer
109 views

Any good practice material for expected value and variance?

I am trying to learn more about probability mass functions, density functions, expected value, and variance. Are there any online materials or quizzes (with answers and explanations) that I can use to ...
4
votes
1answer
66 views

What does “$\Sigma^{\frac12}$” mean in the context of variational auto encoders?

I am studying about Variational Auto Encoders. I found the following equation. $$\mu +\Sigma^\frac12\odot\epsilon$$ I can imagine that $\Sigma$ is related to variance. What does the $\Sigma^{1/2}...
4
votes
1answer
247 views

In the Collatz conjecture, why are $\max(\textrm{collatz}(n))$ and $\textrm{var}(\textrm{collatz}(n))$ so closely related?

Like the question title reads, in the Collatz conjecture, why are $\max(\textrm{collatz}(n))$ and $\textrm{var}(\textrm{collatz}(n))$ so closely related? See the Figure below for a log-log plot. I ...
4
votes
0answers
104 views

The variance of a covariance

The expectation over $x$ of a covariance between variables $A$ and $B$ (where the distribution of $B$ varies according to $x$) is equal to the covariance of $A$ with the expectation of $B$ over $x$: $...
4
votes
0answers
55 views

Seemingly Identical Random Variables with Different Variances

In my probability class, we did the following problem regarding expected values/variance: Consider an experiment where you roll a fair, 6-sided dice until you see a 6. Let $T$ be a random variable ...
3
votes
3answers
1k views

Variance of a sum of IID random variables.

Let's say we have a sequence of $n$ IID random variables, $I_i$. Let's define a new random variable which is their sum: $$S = \sum_i I_i$$ To calculate the variance of $S$ we can say - $$V(S) = \...
3
votes
1answer
5k views

Why are the eigenvalues of a covariance matrix equal to the variance of its eigenvectors?

This assertion came up in a Deep Learning course I am taking. I understand intuitively that the eigenvector with the largest eigenvalue will be the direction in which the most variance occurs. I ...
3
votes
3answers
99 views

Difference between $\operatorname{Var}(Y)$ and $\operatorname{Var}(Y\mid X)$?

What is the difference between $\mathrm{var}(Y)$ and $\mathrm{var}(Y\mid X)$? If $Y = c + \beta X$ and $\operatorname{var}(X)=\sigma^2$, won't both come out to be the same, i.e., $\beta^2\sigma^2$?
3
votes
3answers
1k views

Unbiased estimator of the variance with known population size

The variance is defined as $$\sigma^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^2}{n}$$ where, $\bar x = \frac{\sum_{i=1}^n x_i}{n}$ If someone wants to estimate this parameter from a sample (s)he must ...
3
votes
3answers
93 views

Why is $\operatorname{Var}[X]≤(b−a)^2/4$ if $X$ is a random variable with values between $[a,b]$

I don't know how to start here. I tried to proof it with $$\operatorname{Var}(X) = E(X^2) - [E(X)]^2$$ but this wasn't a good idea in the end.
3
votes
3answers
82 views

We have $20$ students and $15$ lessons. In every lesson, one student is randomly picked and asked a question. Find expected value and variance.

There are $20$ students and $15$ lessons. In each lesson, one student is randomly picked and asked a question by the teacher. Find expected value of amount of students asked a question during the 15 ...
3
votes
3answers
112 views

What's the point of variance?

The big question: Why should I use variance over standard deviation? In what contexts should variance be used (on its own, not with SD)? I'm failing to understand the point of variance - anything I ...
3
votes
3answers
824 views

Why is the denominator $N-p-1$ in estimation of variance?

I was recently going through the book Elements of Statistical Learning by Tibshirani et.al. In this book, while explaining the ordinary least squares model, the authors state that assume that $y_i \...
3
votes
2answers
107 views

Taking the average after randomly knocking out members

I'm looking at a random variable that takes vectors $\newcommand{\vv}{\mathbf{v}} \vv_1, \dotsc, \vv_n \in \mathbb{R}^d$ and calculates their average, after applying "blankout" noise to them. So we ...
3
votes
1answer
173 views

Terms in a stochastic differential equation

In the mathematics of finance, a stochastic process can be given by a stochastic differential equation: $$dX_t = a(X_t,t)dt + b(X_t,t)dB_t$$ where $dB_t$ is a Wiener process. What is the basic reason ...
3
votes
3answers
73 views

Let $X,Y$ be independent RVs. If there exists $c \in \mathbb R $ s.t. $P(X+Y=c)=1$, then $X,Y$ are constants

I would like to show: Let $X,Y$ be independent RVs. If there exists $c \in \mathbb R $ s.t. $P(X+Y=c)=1$, then $X,Y$ are constants a.s.. What I tried: Since X,Y are indep,$$1=P(X+Y=c)=\int 1_{...
3
votes
1answer
112 views

Given a family of probability distributions that are “close to each other” and have expected values 1, 2, 3…, find a lower bound on their variance

Let $\left(\mathcal{D}_{n}\right)_{n\in\mathbb{N}}$ be a family of probability distributions such as: $\forall n\in\mathbb{N}$, $\mathbb{E}\left[\mathcal{D}_{n}\right]=n$: the average value of the n-...