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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37 views

Example value assignments for a discrete random variable X such that E(X) = 1 and Var(X) = 1

I need to define random variables such that their E(X) = 1 and Var(X) = 1 and these values need to be non-negative. So far, the only assignment of values to a random variable that I can think of that ...
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20 views

Taylor Series and Multivariate Delta Method

I'm trying to understand delta method for matrices and vectors to find the variance-covariance matrices for the functions of matrices and vectors. Please see my attempt below. I'm not sure it is right ...
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17 views

Variance from the covariance matrix

I was reading about Common Spatial Pattern. The CSP algorithm tries to find the vector $w^T$ that maximises the ratio of variance between two windows $X_1$ of size $(n,t_1)$ and $X_2$ of size $(n,...
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28 views

Comparing (Normalised) Distribution of Two Small-Sample Datasets

I'd like to compare the distribution of a number of datasets that have few values within them, with the results ideally on a 0-1 normalised scale so that it is clear that a distribution approaching 1 ...
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69 views

Variance of sum of two uniform RV

Let $X$ and $Y$ be two independent random variables, each uniformly distributed on $[-1,1],$ then find $\operatorname{Var}(X+Y).$ My attempt : $$\operatorname{Var}(X+Y) =\operatorname{Var}(X) + \...
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80 views

variance-based upper bound for entropy: proof?

I found the inequality in wikipedia https://en.wikipedia.org/wiki/Entropic_uncertainty $$ H(\phi )\leq \log {\sqrt {2\pi eV(\phi )}}, $$ with $\phi$ as "any probability density function on the real ...
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24 views

Variance of limiting distribution equal to limit of variance

I have a possibly basic question, which I am not sure on whether or not it is true. Suppose we have a sequence of identically distributed, but not necessarily independent random variables $X_n$ on a ...
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26 views

Given that $X_{i+1} = \rho X_{i}$, determine the dispersion matrix $Var[\textbf{X}]$

If $X_{1},X_{2},\ldots,X_{n}$ are random variables and $X_{i+1} = \rho X_{i}$ $(i = 1,2,\ldots,n)$, where $\rho$ is constant, and $\mathrm{Var}[X_1] = \sigma^2$, find $\mathrm{Var}[X]$. MY ATTEMPT ...
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1answer
27 views

Given two random vectors, determine the dispersion matrix $Var[\textbf{X}]$.

Let $\textbf{X} = (X_{1},X_{2},\ldots,X_{n})^{\prime}$ be a vector of random variables, and let $Y_{1} = X_{1}$ and $Y_{i} = X_{i}-X_{i-1}$ where $i = 2,3,\ldots,n$. If $Y_{i}$ are mutually ...
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55 views

Calculate conditional probability using mean and variance [closed]

I have a set where its values follow a normal distribution, but I only have the sum of all of them, and the sum of its squares. Having: $\sum_{i=0}^n X_i$ and $\sum_{i=0}^n X_i^2$. I have the ...
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37 views

Calculate covariance given correlation, problem with percentages

The question is: find the covariance of ABC stock returns with the original portfolio returns. Pretty straightforward. However I get confused working between percentages and units. The ...
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16 views

What is the Variance of the estimator $ \hatμ_n = \frac{1}{n}(5X_1+ \sum_{i=2}^{n-1}X_i-3X_n)$

I am given the random sample X1, . . . , Xn, which is identically and independently distributed over f(X), for a population density f(X) with finite mean μ and variance $σ_2$. I am given the estimator ...
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41 views

Mathematical derivation of why Bagging reduces variance

I am having a problem understanding the following math in derivation that bagging reduces variance. The math is shown but can not work it out as some steps is missing. link
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42 views

Why is $E[(X − b) ^2 ]$ minimal when $b = µ$? [closed]

If $X$ is a random variable with a mean µ and a variance $σ^2$, why is $$E[(X − b) ^2 ]$$ minimal when $b = µ$?
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243 views

Covariance matrix and projection

I have troubles understanding a geometrical meaning of a covariance matrix. Let's say we have a data set containing two points (-1,1), (-1,2) and write them in to the matrix $$D = \begin{bmatrix} -...
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30 views

Hypothesis testing variance using sample mean

I know how to test hypotheses for variance using methods like the chi-square test. However, this problem is asking me to use a rejection region construction in terms of the sum of the sample values (...
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1answer
28 views

Variance of combination of Brownian Motions

Let $Z(t)=W(t)-\frac{t}{T}W(T-t)$ for any $0\leq t\leq T$ with $W(t)$ a Brownian motion, find the variance of $Z(t)$. My attempt: $Var(Z(t))=\mathbb{E}(Z(t)^{2})-\mathbb{E}(Z(t))^{2}$ $Z(t)=W(t)-\...
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Help with a variance proof

I've been doing these exercises, but there is a proof (considering a binomial distribution of $n=4$, where $p$ is the probability of something happening, that has a median $m=4p$, prove, using the ...
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1answer
38 views

Basic Chi square problem

How do I calculate $P(S^2 > 1.8307(\mathrm{PopVariance}))$ if $n =11$? I think I should use the Chi square formula: $$X^2 = \frac{(n-1)s^2}{\mathrm{PopVariance}}$$ But I can't really understand ...
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31 views

How can $\sigma^2$ be derived as a function of $\mu$ in a Gaussian pdf?

I have a Gaussian pdf defined as $$f_X(x) =\frac{1}{\sigma\sqrt{2\pi}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}$$ whose $\mu = \frac{d^2}{6D}$, where $d$ is distance parameter and $D$ is the ...
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Dimensional properties derived from PCA eigenvectors

Background Let's assume I'm using principal component analysis to carry out clustering of a 2-d data set, using a non-normalized covariance matrix to carry out the operation. I then solve for the ...
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29 views

How to calculate variance or diversity index in this chart

The bar shows an information regarding a team, in which 5 people know java, 10 know javascript etc How can I calculate diversity in this team, or perhaps the variance if in this case variance is same ...
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37 views

Are the mean and variance of a set equals the sum of its means and variaces

I'm not so fit in statistics and I found some controverse answers on the internet so I'm asking here. I have a set $A$ with 10439 samples. The set is not of unique values and many of the sample ...
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1answer
21 views

What is the probability mass function of the measured voltage?

In a specific design, the true voltage of a circuit is 250 millivolt(mV). Measurement error that is continuous and uniformly distributed from -3 to +3 mV is added to the true voltage. the measurement ...
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46 views

Is it possible to calculate this special variance?

If i want to estimate the probability, that a random variable $X$ with any continuous distribution takes some value $>a$, i could estimate this with a sample from the correct distribution $X_1,...,...
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165 views

Variance of a Brownian motion

Let $\{X(t), t \geq 0\}$ be a Brownian motion with drift parameter $\mu = 3$ and variance parameter $\sigma^2 = 9$. If $X(0) = 10$, find $P(X(0.5) > 10)$. First, I calculated the expectation and ...
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51 views

Scalar product induced by covariance matrix

Suppose that $n$-dimensional random vector $Y$ has covariance matrix $\Sigma$. It is well known that for any $a\in\mathbb{R}^n$ we have \begin{align} var(a^TY)=a^T\Sigma a. \end{align} Is there any ...
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22 views

Variance and covariances from linear mixed model for power simulation using R

I am working with longitudinal data where the outcome is the number of steps per minute. My LMM fit would look like: ...
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120 views

Variance of sum of $m$ dependent random variables

Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random variables with $\mathbb{E}[X_i]=0$, $0<Var(X_i)<\infty$ ($m$-dependent means that each $X_i$ is independent of ...
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44 views

convergence almost surely implies convergence of variance?

Does $X_{n} \to X$ almost surely imply that $Var[X_{n}] \to Var[X]?$ I saw this post convergence in mean square implies convergence of variance which states that $X_{n} \to X$ in $L^{2}$ implies ...
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Adding variances of a variable

I have the average variance of a stock's daily return over a year and I want to know whether or not I can times this number by 252 (number of trading days in a year) to find the yearly variance. My ...
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5 views

Variance estimation of a secondary, non-observable variable

Sorry for lack of math rigor in advance. I want to estimate variance of a stochastic variable $Y$ that is only observable through a continuous, time limited function $f(t)$ as $f(t+Y)$ Regarding $f(...
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49 views

Finding variance without the data

A researcher is testing the effectiveness of a political video. She has randomly sampled 120 people to test the video. For each person, there is 50% chance the person will be Democrat and 50% chance ...
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1answer
31 views

Regression model + expected value, variance and autocorrelation of the error term

Consider this regression model $$Y_t=X_t\beta+\epsilon_t, ~~~~~~~~~~\epsilon_t \sim WN(0, \sigma^2_{\epsilon})$$ with 3 different specifications of the error term: $\epsilon_t=\alpha_1\epsilon_{t-1}+...
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1answer
18 views

MLE : Effect of incorrect variance on the mean for a normal distribution

Consider we have univariate samples, $x_k$, belonging to a category $\omega$ drawn from a dataset D according to an assumed distribution $p(x|\omega)$ $\sim N(\mu, 1)$. However, let the true ...
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23 views

Variance of truncated 2d Gaussian

To find the expectation of the Truncated Gaussian E($z_1^2$| $z_1^2 \leq \tau , z_2^2 \geq \tau$). Where $\boldsymbol{z} = [z_1,z_2]^T$ and $\boldsymbol{z} \sim \mathcal{N}(\boldsymbol{0},C)$, where $...
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41 views

Variance of double integral of changing error

I have the following case and would like to know if there is a better approach than the one I am currently following. I have an array of errors ($\epsilon(i)$) and an array of corresponding variances ...
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1answer
25 views

Conditional Expectation of a Sum of Random Variables and a Random Integer

Let $(X_n : n \in \Bbb N)$ be a sequence of identically distributed random variables, with mean $ \mu$ and variance $\sigma^2 < \infty$. Set $S_0 = 0$ and $S_n=X_1+X_2+...+X_n$ for $n>0$. Let $N$...
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27 views

What is the minimum variance band of Poisson Distribution?

I am trying to calculate the minimum variance bound of Poisson Distribution. poisson distribution: P(X=x)=(λ^x)/x! e^-λ, were λ is the mean. I got λ/(sum of x), but I am not sure if this is right. ...
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1answer
30 views

How to show that the error variance of the best linear predictor is inferior to the proportional predictor?

Let's consider the 1D case. How do we prove that the error variance of the Best Linear Predictor (BLP) is inferior than the Proportional Predictor (i.e. the Linear Predictor without the intercept)? ...
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2answers
90 views

About Conditional Variance $X$ Has distribution $ U(0,1)$ and $Y$ has distribution $ U(0,X)$

I have a question about the problem mentioned above, the main says $X$ Has distribution $ U(0,1)$ and $Y$ has distribution $ U(0,X)$ Find $E(Y)$ and $Var(Y)$ I try to take it for $E[Y|X]=X$ and $...
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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 ...
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88 views

Show that T achieves the Cramer Rao lower bound

Problem Statement: Consider $T$ to be an estimator of $\theta$. Show that $T$ achieves the Cramer Rao lower bound if and only if $Z$ is a linear function of $T$ $Z=a(\theta)T+b(\theta)$ ...
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1answer
123 views

We have an urn with 6 red balls and 4 green balls.

We have an urn with 6 red balls and 4 green balls. We draw balls from the urn one by one without replacement, noting the order of the colors, until the urn is empty. Let X be the number of red balls ...
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1answer
14 views

$Y = \frac { K A ^ { 3 } } { ( B + D ) ( C - D ) }$

K is a constant Find an expression to approximately determine the variance of Y, assuming $A , B , C ,$ and $D$ are probabilistically independent. isnt the expression they have already given me the ...
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1answer
46 views

Minimum variance of $k_1X+k_2Y$ where $X,Y$ are independent Poisson

I have the following question for homework: Suppose that $X$ and $Y$ are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the combined estimator ...
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2answers
204 views

Variance and mean of balls in bins limited capacity

Let there be $m$ indistinguishable balls, $k$ bins, $C$ capacity. Let $X_j$ denote the total balls in bin $j$. I've seen ways to calculate the total number of combinations, but I'm not sure how to go ...
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26 views

A random variable $X$ is number of boys out of $n$ children. Calculate $\operatorname{Var}(2X-n)$

Let a random variable $X$ be the number of boys out of $n$ children. The probability to have a boy or a girl is $0.5$. Calculate $V(2X-n)$. I know that $Var(2X-n)=4V(X)$. $\mathbb{P}(X=k)={1\over 2^...
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2answers
161 views

Formula to recalculate Variance after removing a value and adding another one given old variance

Let's say I have a data set of $10,20,30$. My mean and variance here are mean= $20$ and variance = $66.667$. Is there a formula that lets me calculate the new variance value if I was to remove $10$ ...
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1answer
23 views

Partial permutation of time sequence data that keep order of events

Suppose you have sequence S of N elements that are descending ordered by time. How many ways can you take K element subsets from S preserving time descending ordering? example for sequence S={A,B,C,D,...