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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3 views

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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33 views

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 ...
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27 views

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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2answers
25 views

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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8 views

Right expression for the computation of a Fisher matrix element

I have tried to calculate the expression of a Fisher matrix element $(i,j)$ : First method : with discrete values I get the following definition of an element $(i, j)$ of Fisher matrix $F$ : $$ F_{i ...
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41 views

why least squares estimates are unbiased? [closed]

I'm reading ESE (Element of Statistical Learning) and i'm struggling with this part: 1 My questions are: why $ε$ is proportional to $ N(0,σ^2) $ ? can you explain the second part of the text ...
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1answer
18 views

The conditional mean affect variance and covariance

From this dicussion, $$ \hat \beta = \frac{\widehat{\mathrm{cov}}(y_i, x_i)}{\widehat{\mathrm{var}}(x_i)}. \tag{3} $$ $$ \hat \gamma = \frac{\frac{1}{n} \sum_i y_i x_i}{\frac{1}{n} \sum_i (x_i)^2} \...
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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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67 views

Converting a normal random variable

According to what I have learnt, a random variable, X, with mean, $\mu$ and standard deviation, $\sigma$, can be converted to the standardised variable, Z, using the formula: $$Z = \frac{X-\mu}{\sigma}...
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24 views

Variance of a complicated equation [closed]

I have an equation $$ Z|\mathcal{H_i}=\sum_{n=1}^{N}\left(|h_i|^2|s[n]|^2 +|w[n]|^2 +2\Re(h_is[n]w^{*}[n])\right), = \sum_{n=1}^{N} Z_n|\mathcal{H_i} $$ where $s[n]\sim\mathcal{CN}(0,P_s),h_i\sim\...
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1answer
40 views

Comparing mean squared errors for estimators

Let random sample $(X_{1},...X_{n})$ is taken from a population with mean $\mu $ and variance $\sigma ^{2}$. Compare suggesting estimators for $\mu $ according to mean squared error. Suggesting ...
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26 views

How to calculate the variance of a two-particle distribution function?

I have calculated a two-particle distribution function called $f(r)$ where $r$ is the distance between of two particle (also there are three parameters,$\alpha$, $\beta$ and $\gamma$ in its formula ...
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1answer
28 views

How to find a point estimate for a given random sample of exponential distribution given the sample variance and four out of five sample values?

Let $\left(x_1,x_2,x_3,x_4,x_5\right)$ be the observed values of a random sample of size $5$ from an exponential distribution with parameter $\beta$. Out of five observed values four are given as $x_1=...
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A professor knows that the test score of a student taking her exam is a random variable with mean $75$ and variance $25$. Find the $n$ [closed]

A professor knows that the test score of a student taking her exam is a random variable with mean $75$ and variance $25$. Find the $n$, the number of students to ensure that with probability $0.9375$ ...
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1answer
143 views

Mean and variance of a binomial random variable wrt $n$ when $p$ is a function of $n$

I would appreciate any suggestions regarding the following. For a constant $c>0$: $$f(n,p;\,c):=\sum_{k=1}^{n}\binom{n-1}{k-1}(p)^{k-1}(1-p)^{n-k}\cdot\frac{1}{k}-c$$ and let $p^{*}=p^{*}(n;\,c)$ ...
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1answer
25 views

Variance of random walk

Consider the Random Walk $$X_t = X_{t-1}+\epsilon_t$$ with $\epsilon_t \sim N(\mu,\sigma^2)$ and $X_0=0$. We can write $$X_t=X_0+\sum_{n=1}^t\epsilon_t.$$ Using the equation above we have $$\mathbb{E}[...
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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 $X$ ...
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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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27 views

Question that has equalities to prove about variance.

Let $X,Y$ be two discrete random variables with finite expected value and variance, we define: $Var(Y|X=x)=E[(Y-E(Y|X=x))^2|X=x]$ for $x$ such that $P_X(x)>0$. The variance of $Y$ given $X$: $Var(...
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17 views

Variance of a shifted exponential

I am having trouble finding the variance of this distribution. the mean is 2$\lambda$, but the variance is $\lambda^2$ ? In the solution, they say $\operatorname{Var}(\bar Y) =\lambda^2 $, I'm really ...
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Unbiased estimator and its variance on weighted sum of the population

Say we have a population of $N$ units, $\{x_1,\cdots,x_N\}$, where $x_i\in[0,1]$. For each $x_i$, we know its probability of being selected as a single random sample, denoted as $\pi_i$. Given $N$ is ...
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2answers
38 views

Calculating the variance using the Probability Mass Function (PMF)

My Problem: The random variables X and Y have the simultaneous probability function pX,Y(x, y). pX,Y(x, y) X = 0 X = 1 X = 2 Y = 0 0.1 0.2 0.2 Y = 1 0.1 0.1 0.3 What is Variance(X+Y) ? Answer: The ...
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Linear Relationship for Bias Adjustment Term

I have written a post that outlines Air Bnb's adjustment for winner's curse - blog, paper. The bias term in figure 2 is said to increase linearly with delta (see blog or paper link, figure 1). However,...
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How can I cope with asymmetry when looking at forecast vs actual errors to calculate a fair measure of variation?

I have a data set that contains an actual value and two different forecasts for each datapoint. I want to compare the quality of each forecasting method. I started by drawing a histogram of relative ...
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113 views

Largest eigenvalue of covariance matrix

Consider a random vector $X\in\mathbb{R}^d$ that is distributed according to a multivariate Gaussian $\mathcal{N}(\mu, \Sigma)$, and the random variable $y = \langle a, X \rangle + e$ for some (fixed) ...
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Maximum likelihood estimation for variance from the data

For the maximum-likelihood, I have set $l(\theta) = \sum_{k=1}^{n} ln (p(x_{k}|\theta))$ and $\Delta_{\theta}l(\theta) = \sum_{k=1}^{n} ln (p(x_{k}|\theta))$ Suppose I have a log probability $p(x_{k}|\...
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21 views

proof related to expected values

I wanna prove following expression. $$ n\sum_{i=1}^{n} {X_i^2} - (\sum_{i=1}^{n}{X_i})^2 = n\sum_{i=1}^{n}{(X_i-\bar X)^2}$$ pf) $$ n\sum_{i=1}^{n} {X_i^2} - (\sum_{i=1}^{n}{X_i})^2 = n\sum_{i=1}^{n} {...
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26 views

Variance of a function g(x)=(2x+1)^2 where x is random variable of values 3 6 9 with probabilities 1/6 1/2 and 1/3 respectively [closed]

Solving for $Var[g(x)]$ by computing $E[g(x)^2]-(E[g(x)])^2$ gives the answer: [] While solving for variance by breaking down the g(x) into individual values gives $Var[g(x)] = Var[(2x+1)^2]$ $Var[(4x^...
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1answer
45 views

Probability question with expected value and variance. (Involves Markov and Chebyshev's inequalities)

A man bought $4$ stocks, the value of the first stock is $100\$ $ , the second $150\$$, third $50\$$ and the fourth $20\$$. We define $v_i$ to be the value of the stock $i$. The probability for the ...
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Coefficient of variation of circular data

The definition of coefficient of variation is as follows: coefficient of variation= standard deviation/mean. I am using circular (directional) statistics to find the mean and standard deviation. My ...
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1answer
17 views

How to find MSE based on observations from 2 samples.

I have 2 independent random samples: {$X_1, X_2$} and {$Y_1, Y_2...Y_{22}$} with E[$X_i$] = $\mu$ and E[$Y_j$] = $3\mu$, Var[$X_i$]= $\sigma^2$ and Var[$Y_j$]= $\sigma^2/2$. Now I have the following ...
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43 views

Does Bayes theorem hold for variance?

$\DeclareMathOperator{\Var}{Var}$ I am trying to find $\Var[A\mid B]$ of a random walk when $A < B$. I was wondering whether i can apply the Bayes theorem which says that $P[A\mid B]$ = $\dfrac{P[...
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28 views

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 ...
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1answer
18 views

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 ...
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1answer
35 views

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 ...
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1answer
24 views

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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“Reverse” Chebyshev Inequality that gives lower bound of being far from mean [closed]

Chebyshev's inequality gives an upper bound on $P(|X - \mu| \geq k\sigma)$ but I was wondering if there was a way to find a lower bound for this probability or, equivalently an upper bound on $P(|X - \...
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27 views

How to quickly find the variance of a Gaussian random variable from the density?

The normal density looks like $f(x) \propto \exp\left( -\frac{(x-\mu)^2}{2\sigma^2} \right)$. Suppose that you know that $X \sim N(\cdot, \cdot)$ is normal, but the density looks like $f_X(x) \propto \...
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35 views

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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How to show that $\operatorname{var}(X_{(1)}) = \operatorname{var}(X_{(n)})$ for $X_i \sim U(0, 1)$?

Define $Y_i = 1 - X_i$ and $Y_{(k)}$ as the k-th order statistic, then $$ Y_{(1)} = 1 - X_{(n)} $$ Variance is invariant to shifts, so $$ \operatorname{var}(X_{(n)}) = \operatorname{var}(Y_{(1)}) \\ \...
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22 views

Show that $\text{Cov}(AY)=A\text{Cov}(Y)A^T$

Let $Y$ be a $p$-dimensional random vector $\mathbb{E}|Y|^2\lt \infty$, and $A$ be a $q\times p$ deterministic matrix. Show that $$\text{Cov}(AY)=A\text{Cov}(Y)A^T$$ My solution: $$\text{Cov}(AY)\...
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1answer
19 views

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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1answer
49 views

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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3answers
77 views

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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13 views

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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3answers
41 views

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 ...

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