# 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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### Find vector X that minimizes variance of the vector |AX|^2 where A is a matrix

I have a complex matrix $A$ of dimension $M$x$N$ which is known. I am now looking for a complex column vector $X$ of dimension $N$x$1$ to do the multiplication $AX=Y$, which will be a new column ...
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### What is the logic behind converting these units of standard deviation?

I was solving a question from high school, and it was asking for a conversion of units of one given standard deviation (σ). But I really didn't get why it is logical just converting this way without ...
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### Why does the empirical standard deviation $\hat{\sigma}$ satisfy $E\hat{\sigma} = E\lvert X^{i}-E[X^{i}]\rvert$

Let $(X^{i})_{i=1,...,N}$ be iid random variables. Why does the empirical standard deviation $\hat{\sigma}$ satisfy $E\hat{\sigma} = E\lvert X^{i}-E[X^{i}]\rvert$? The empirical standard deviation is ...
1answer
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### What the hint to find the probability if given mean and variance?

The amount of time needed for a printer to print a file is a random variable with mean $E(X_i)=20$ minutes and variance $var(X_i)=4$ minutes$^2$. The times needed for difference file are independent ...
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### How can I calculate the covariance of 2 random variables, given the second one and the variance of the first one?

If X is a random variable with variance 1 and $Y = -2X+5$ how do I calculate the covariance of X and Y? I know the formula of the covariance is $cov(X,Y) = E(XY) - E(X)E(Y)$, but from the given data, ...
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### How to “normalize” variance of $Y = A X$? [closed]

Assume that $X$ has identity variance matrix. How to "normalize" variance of $Y = A X$? I know that Y is a symmetric real matrix, also I knwo the fact that Var(A X) = A Var (X) A^T. Var(X) = ...
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### Central limit theorem about additive white Gaussian noise (AWGN)

I came across reading material in which AWGN is assumed as $\mathcal{C}\mathcal{N}\sim (0,N_w)$......(1) I understood $(1)$ clearly. But later it is mentioned that "As per central limit theorem (...
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### MSE of Ridge estimator, Linear Regression.

I have the following expressions: $$Bias (\hat\beta_{ridge}) = ((X^TX + \lambda I)^{-1}X^TX-I)\beta$$ $$Var(\hat\beta_{ridge}) = \sigma^2(X^TX+\lambda I)^{-1}X^TX(X^TX+\lambda I)^{-1}$$ where $X$ ...
1answer
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### Finding expectation using iterated expectation in a production line case

A factory produces bolts with a defective rate that changes randomly and independently from day to day but is constant throughout any given day. Let $p_i$ denote the defective rate on day i, and ...
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### Unidimensional variability measure for multivariate random samples (or time series)

I have multiple samples on $n$-dimensional random vectors (from a time-series). I'll like to have a unidimensional measure of its variability. A natural one (discussed here) is to extend the variance ...
1answer
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### Minimum variance for a given IQR in a sample of 7

for a sample of 7 elements it is given that its inter-quartile range $Q3-Q1$ equals $4$. I need to find the minimum value the variance can take. No other information about the sample is given. ...
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### Calculating “fairness” in a Handicap Formula

I run a website dedicated to golf in which the user can configure their handicap formula with their own settings. What I would like to do is create a graph or a value rating (maybe from 0 - 100) for ...
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### Variance of the Square

Suppose $X_1, \cdots, X_n$ are a sample of independent variables taken from a normally distributed population with mean $\mu$ and variance $\sigma^2$. I would like to determine the variance of the ...
1answer
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### Remove half the points to minimize variance

I have a set of $n$ points in $\mathbb{R}^d$ and I'm trying to find the subset of $\frac{n}{2}$ points with the smallest variance. It can be shown that there exists a point in the set such that if we ...
1answer
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### Adding Variances vs Not

The problem I have with this is calculating the variance of the weight of the 15 books. To me, X is the RV for the weight of 1 book, where the mean = 12, and variance = 15 (root 15 squared). Let Y be ...
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### Calculate mean and variance of a function of random variables

I am working on a problem and I need to compute the mean and variance of $Y$, i.e. $E\{Y\}$ and $E\{Y^{2}\}$ is required, where $$Y = \frac{A^{2}+B^{2}+AB+CD}{\sqrt{(A+B)^{2}+(C+D)^{2}}},$$ where A, ...
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### Cramer-Rao lower bound for an efficient estimator

Let's assume $x\left(n\right)\:=\:b^n+w\left(n\right)$ where $w\left(n\right)$ has a normal distribution of $w\left(n\right)\in N\left(0,\sigma ^2\right)$ I need to estimate $b$ by finding the CRLB (...
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### Find $Var(XY)$ for $X,Y$ chosen from a unit square.

Let $(X, Y)$ be a point chosen at random on the unit square $[0, 1] × [0, 1]$. Find $Var[XY]$. My Attempt (I think I have the right answer. I just want some verification. Thank you) First, we want to ...
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### How can I construct a PDF that has infinite variance?

I want to construct a PDF that has infinite variance. So I started with the definition of variance $$\operatorname{var}(X) = E[X^2] - E[X]^2$$ I'll constraint the problem to a symmetric distribution ...
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### Bounded variance for Lipschitz function of random variable

In Priors for Infinite Networks (Neal, 1996), part of the proof is that $\tanh(X)$ for Gaussian RV $X$ has finite variance, which is later used for the Central Limit Theorem. For arbitrary activation ...
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### Expected value and variance of a set of random variables

Suppose $X_1, X_2, \ldots , X_n$ are $n$ independent r.v.s, with the same probability distribution and with mean $\mu$ and variance $\sigma^2$. Let $$\bar{X}=\frac{X_1+X_2+\cdots+X_n}{n}$$ I know ...