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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Linear Combinations of Sample Mean Difference

What is the standard error of the mean difference between 2 variables (a & b)? I have the following data: $$ \sigma_{A} = 1.813529 \nonumber \\ \sigma_{B} = 1.932183 \nonumber \\ cov(A,B) = 2....
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How to find the variance of a biased estimator of variance of standard normal distributed random variable?

Let $Y_i$ be an independently and identically distributed standard normal random variable. Denote $S_1^2$ as estimator of $\text{Var}(Y_i)$. How to find the variance of the biased estimator $S_1^2$? $$...
Ermaolaoye's user avatar
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How to prove the inequality for the standard deviation of a linear combination of two random variables

The question comes from ‘Mathematics for Finance: An Introduction to Financial Engineering’ by Marek Capiński (Author), Tomasz Zastawniak. The book provides a conclusion about the risk and return of ...
bokabokaboka's user avatar
2 votes
1 answer
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Does the kurtosis need to be finite for the sample variance to be consistent?

It is known (see this answer) that if $\mu_4$ is the fourth central moment of a distribution and $\sigma$ is the standard deviation, then we can write $$\operatorname{Var}(S^2_n)=\frac{1}{n}\left[\...
harrydiv321's user avatar
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Probability: Question about expected value with coins

In a coin purse, there are two one-dollar coins and one five-dollar coin. One randomly draws without replacement two coins and sets X = the value of the first coin, Y = the value of the second coin. ...
Need_MathHelp's user avatar
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Calculating Variance in Waiting Time for a Queueing Network

I'm working on a queueing network model that incorporates blocking and features two states. After defining the global balance equations, I solved them for my parameters arrival rates (λ), service ...
RookieScientist's user avatar
2 votes
1 answer
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Where am I wrong in this variance calculation?

It is said that if $X$ and $Y$ are independent, then $\operatorname{Var}[X+Y]=\operatorname{Var}[X]+\operatorname{Var}[Y]$. Given that $$Var[X]=E[X^{2}]-E^{2}[X]$$ then by substitution we arrive at $$...
excitedGoose's user avatar
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Variance for a Piecewise Function

Let $X_1,X_2,...,X_n$ be independent and identically distributed continuous random variables with the probability density function (p.d.f.) $ f(x| \mu)= \begin{cases} \frac{1}{\mu}e^{-\frac{-x}{\mu}}&...
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Variance of affine transformation of random variable

I am trying to understand how to derive the variance of an affine transformation of a random variable $X$. In particular, I don't understand the following result: $$\mathbb{V}_X[Ax]=A\mathbb{V}_x[x]A^...
gc5's user avatar
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Suppose $X$ and $Y$ are independent. What is $Corr(X+Y, X-Y)$? [closed]

Let $X$ and $Y$ be independent with $V(X) = 25, V(Y)= 9$. Find $Corr(X+Y, X-Y)$. Is it correct that $Corr(X+Y, X-Y) = Cov(X,X) - Cov(X,Y) + Cov(Y,X) - Cov(Y,Y) = Var(X) + Var(Y) = 25 + 9 = 34$?
Sha's user avatar
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Variance of rounded standard normal random variable

I would like to evaluate the variance of $y=\mathrm{round}\,x$, where $x\sim\mathsf{Normal}(0,1)$. Using numerical integration, $\mathrm{var}\,y\approx 1.0833333223611180232$ which is very close to $...
Till Hoffmann's user avatar
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PMF of Sum of Values Selected Without Replacement

I’m currently stuck on this question. You have $m$ values, $X_1, X_2, …, X_m$ where each value is selected without replacement from the range $[1, m+n]$ inclusive without replacement. $Y = X_1+ X_2+ \...
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How to calculate historical returns and variance for a single asset with non buy-and-hold strategy

Suppose i have a strategy that is not buy-and-hold type of strategy. It can have unique entry timing and unique exit timing for a single asset and both long, and short positions will be allowed, and ...
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How to use ICC with given data [migrated]

I have data with columns like this: ENTITY Avg_Score_1 N_obs_1 Avg_Score_2 N_obs_2 With sample values: 001 | 0.997 | 900 | 1.13 | 905 002 | 0.890 | 250 | 0.96 | 251 For about 1000 unique ENTITY values,...
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Variance of a function with two random variables

I have a function $f$ which depends on two random variables $a$ and $y$: $f(x,y)$. I want to calculate the variance of $f$ w.r.t. $x$ and $y$, i.e. $\mathrm{Var}(f(x,y))$. I am wondering if I can ...
Shashank 's user avatar
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Building random variable with given variance and expectation

I'm given variance V, expectation E and number n - amount of values random variable takes. ...
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Kernel, variance and estimation

In the paper by Terrell 1990, in the Theorem 1 below on the page 471, I would like to derive the formulas for $g(x)$ and $h(x)$ and perhaps also why $\beta(k+2,k+2)$ minimizes that integral given.Why ...
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Law of total variance on function of random variables

Let's take the following random variables $X,Y,Z_1,...,Z_K$, on the same probability space (or associated with the same experiment), where we know that $$ Y = \sum_{k=1}^K\beta_kZ_k$$ Let's say I am ...
BMBE's user avatar
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Difference between random variable $2X$ and the sum of two independent observations of $X$? [closed]

Consider this Dice A has the numbers 1, 1, 2, 2, 2, 4. We find the expected value to be $E(X)=2$ and the $Var(X)=1$. Dice B has the numbers 2, 2, 4, 4, 4, 8. As one would expect, $E(2X)=4$ and the $...
Quin Gardiner Bax's user avatar
1 vote
1 answer
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Law of total variance on conditional expectation

Let's take 3 random variables $X,Y,Z$ on the same probability space (or associated with the same experiment), then the law of total variance states that: $$ V[X] = V[E[X|Z]] + E[V[X|Z]] $$ Then what ...
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How do I go from a normal distribution with a variance to uniform distribution with the same variance?

you have some normal distribution with mean 0 and variance $𝜎^2$ and that you want the interval [𝑎,𝑏] such that the uniform distribution on the interval has mean 0 and variance $𝜎^2$ ? This is my ...
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What does it mean for the variance to be squared?

Suppose we are measuring the heights of different people. Our measures will be in $cm$, but the variance will be in $cm^2$. If we were measuring the time to complete a race course, our measures would ...
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Find $k$ using Chebyshev's inequality such that $E(X)=100,\sigma(X)=10$ and $P(X\le 10k)\ge 4/5$

Given $E(X)=100,\sigma(X)=10$ and $P(X\le 10k)\ge 4/5$, how do we find a lower bound on $k$ using the Chebyshev's inequality Let $X$ be any r.v. with finite mean, $μ$, and finite variance. Then $∀a &...
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Fisher Information and Parameter Space

I am reviewing Fisher information and saw that one of the requirements is that the distribution of the data, say $f(x|\theta)$, involves a parameter $\theta$ that is unknown but lies within a given ...
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Variance of sum of products of normally distributed random variables?

I want to compute $Var[A+B+C]$ where $A$, $B$, and $C$ are not independent of each other. In particular, I don't know how to compute $Cov[A,B]$, $Cov[A,C]$, and $Cov[B,C]$. The model specifications ...
anonymous 's user avatar
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Variance at a single measurement

I have a collection of data points, $(x_0, y_0)...(x_{n-1}, y_{n-1})$ of the function $y = y(x)$ where the values of x are in ascending order. I'm working out an algorithm for smoothing splines, ...
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1 answer
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Proof that sample variance is biased in presence of autocorrelation

With no correlation we can show sample variance is unbiased: $$E[s^2] = E\left(\frac{\sum^n_{i=1}(X_i - \bar{X})^2}{n-1}\right) = \sigma^2$$ Proof $$E\left(\sum^n_{i=1}(X_i - \bar{X})^2\right) = E\...
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Sampling variance of edge density of subgraphs

I would like to evaluate the mean and variance of the edge density for subgraphs obtained by repeatedly subsampling nodes. Specifically, suppose we have an undirected graph $G$ with $N$ vertices and ...
Till Hoffmann's user avatar
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1 answer
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Relationship between variance and covariance

I know that $$\text{var}(x-y) = \text{var}(x) + \text{var}(y) - 2\text{cov}(x,y)$$ and $$\text{cov}(x,y) = \frac{1}{2}(\text{var}(x) + \text{var}(y) - \text{var}(x-y)).$$ Is it possible to say that $$\...
lela's user avatar
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Why is the residual variance / pooled sample variance divided by n-k in ANOVA?

I was looking for a proof such as for sample variance where it's shown that expected value of sample variance with n-1 in the denominator yields the parameter. I'm not even sure what pooled sample ...
Maciej Jałocha's user avatar
2 votes
1 answer
48 views

What is the intuition behind the single-pass algorithm (Welford's method) for the corrected sum of squares?

The corrected sum of squares is the sum of squares of the deviations of a set of values about its mean. $$ S = \sum_{i=1}^k\space\space(x_i - \bar x)^2 $$ We can calculate the mean in a streaming ...
Foobar's user avatar
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Asymptotic distribution of estimator of the estimator of the standard deviation

I have the following task: Let $Y$ be a binomial random variable. Find the asymptotic distribution of the ML estimate and find the asymptotic distribution of the estimator $(p(1-p))^{\frac{1}{2}}$ of ...
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2 votes
2 answers
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An exercise question in the probability theory class

I was asked a "simple" question by one of my students in the tutorial class, but I found myself struggling with this for about 2 hours already. Here is the question: Assume there are $K$ ...
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Incompatible results in finding the variance of random variable

Let $f(x) = 3/64(2x - 2)$ whenever $x \in [1, 5]$, zero otherwise, be the PDF of a random var. I know the expected value of $X$ is $2.75$. I was asked to find its variance. For the sake of practice I ...
lafinur's user avatar
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Computing the variance of $X \wedge (n-X)$ where $X\sim \text{Binomial}(n,p)$

I want to calculate variance of $Y=X \wedge (n-X)$ where $X\sim \text{Binomial}(n,p)$ and $n$ is even. I have managed to determine the distribution of $X \wedge (n-X)$. We have $$\mathbb{P}(Y=k) = 2\...
Snildt's user avatar
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Intuition Behind Xavier Initialization

Can anyone explain why to compromise between the variances 1/n_in and 1/n_out its 2/(n_in+n_out) its not an average or a harmonic mean it seems random to me? The full paper is here:https://proceedings....
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Prove or give counter example of quadratic inequality

I have two finite probability mass functions (pmfs) $P(x)$ and $Q(x)$ on the same support $(0,1,\ldots,n)$. Let $(p_0,p_1,\ldots,p_n)$ and $(q_0,q_1,\ldots,q_n)$ be the probability vectors from the ...
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Relationship between model variance and dataset size for SGD

I'm looking for some function that describes the variance of model weights when trained with Stochastic Gradient Descent for $m$ independent minibatches. I can apply central limit theorem to a single ...
joseph rance's user avatar
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31 views

Expectation and variance of bivariate skew normal distribution

I am fitting a bivariate skew normal distribution to a 2D data through the sn package in R. I get a $2 \times 1$ vector of ...
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How to Calculate the Average Variance of P_{S|XZ} with Respect to P_{Z|X}?

Question: Calculating Average Variance in a Gaussian Mixture Model I am working with a Gaussian Mixture (GM) model within the framework of the Expectation-Maximization (EM) algorithm. My goal is to ...
Alireza Ghazavi's user avatar
1 vote
0 answers
57 views

Variance of the sum of squared binomial distributed variables, where the total sum is a constant

The problem Let $X \sim Binomial(l, p)$ and $\sum_{i=1}^{m} X_i = n$, where $l$, $m$, $n$ and $p$ are all known constants. Find: $$Var\left(\sum_{i=1}^{m} X_i^2 \right)$$ What I tried My first attempt ...
chipmunk's user avatar
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24 views

Covariance of Best Linear Unbiased Estimators and arbitrary LUE

I'm working on a problem involving two linear unbiased estimators $T$ and $T'$ of a parameter $\theta$, defined from a sample $\{X_1, \dots, X_n\}$ with mean $\theta$ and finite variance. I aim to ...
Taha Rhaouti's user avatar
1 vote
1 answer
62 views

Number of paths with vertices of the same color in a binary tree with random coloring of vertices

Question from an old exam: In a full binary tree of height $n$, each vertex is randomly colored either white or black. A path is considered good if it goes from any vertex to a leaf and all vertices ...
Michał's user avatar
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Orthogonal transformation of Heteroskedastic matrices

Consider two $N \times N$ dimensional real matrices $A$ and $B$. $A$ is a diagonal matrix with all non-zero elements taken from a real Gaussian distribution with mean $\mu = 0$ and variance $\sigma = \...
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Variance of Multimodal Generalized von Mises Distribution

I'm working with the Multimodal Generalized von Mises (MGvM) distribution, and I am interested in finding the variance of this distribution. The density function is given by: $$ f(\theta) = G_M(\mu_1, ...
Alireza Ghazavi's user avatar
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40 views

Convex range, peak to peak, or deviation function

I have a non-negative function $f(a,x)\geq 0$ which is convex in its second argument, such that $$f(a,\lambda x_{1}+(1-\lambda)x_{2})\leq \lambda f(a,x_{1})+(1-\lambda)f(a,x_{2})$$ for every $a$ and $\...
Guy Ohayon's user avatar
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59 views

Given N observations - Bayesian Posterior for Unknown Variance of a Normal Distribution with a Known Mean?

So, starting from no information besides N trials from a Gaussian with $\mu = 0$, I'd like to know the best Bayesian posterior for the unknown variance, $\sigma^2$. My approach so far as been to ...
SSD's user avatar
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1 answer
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Calculate the Variance of $\min(N_k,p)$

I am trying to compute the variance of a random variable $\min(N_k,p)$, where $N_k$ is a random variable and $p$ is a fixed number. I have computed the expectation ...
Sumit Singh's user avatar
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1 answer
68 views

Why is the variance is squared? $V(R) = α^2V(R_A) + β^2V(R_B) + γ^2V(R_C)$ [closed]

When considering the variance of a portfolio with the returns of three assets $(R_A, R_B, R_C)$ and their respective weights $(\alpha, \beta, \gamma)$, the equation is given as: $$ V(R) = \alpha^2 V(...
Kanu's user avatar
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Why can't we rule out negative variances when using gerber statistic for creating the covariance matrix?

I recently read that one problem with gerber statistic is that we cannot guarantee the statistic matrix to be positive semi-definite. Why is that the case and how is it different from normal ...
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