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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.

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maximum sum of two subset standard deviation

Given a finite set defined in real number $\{x \mid 0< x < 1 \}$ with size $N > 3$ If I split $\{x\}$ into two subsets $\{A\}$ and $\{B\}$ , $A\cup B = X$ and $A\cap B = \emptyset $, How ...
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Proving the Linearity of Expectation from the opposite side

An alternative method of variance is $$E[X^2]-(E[X])^2$$ and proving that $E[(X-E[X])^2] = E[X^2]-(E[X])^2$ is done. But how about the other way around $E[X^2]-(E[X])^2 = E[(X-E[X])^2]$, how can you ...
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19 views

$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$

We know that the conditional variance of a multivariate normal vector $(X,Y)$ is equal to the Schur complement: $$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$$ However, $\Sigma_{XX}-\...
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Minimizing the sum of absolute difference of each point from the average does it minimize the variance?

I have a problem in which there is a set of $N$ points. Each point has a set of possible weight (lets call them $W_i$ where $i \in [1,N])$. My goal is to select the correct weight $W_i$ for each point ...
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Solomon four group design

I'm trying to calculate a solomon four group design in rstudio. The literature with the executation steps are here. But I'm bit confused about the first step on page 4, perform 2 x 2 ANOVA. What ...
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2answers
63 views

Tip: Chebyshev Inequality for $n$ throws of a dice

Say a fair dice is thrown $n$ times. Showing using the Chebyshev Inequality that the probability that the number of sixes thrown lies between $\frac{1}{6}n-\sqrt{n}$ and $\frac{1}{6}n+\sqrt{n}$ is at ...
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Statistics Calculate the variance of the data if 156 is added. [closed]

If variance is 18 of 5 data whose mean is 150 then what will variance if 156 is added on data?
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18 views

Subset variance order preserving function

Given a finite set with real numbers. X = {x1, x2, x3}. There can be a unique order defined for all the subsets using Variance operator. e.g. X = {1, 2, 4}. $$ {\displaystyle \operatorname {Var} (X)...
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32 views

How to compute variance of conditional expectation?

Suppose that $(\xi, \eta)$ is a random vector with absolute continuous distribution: $$f_{(\xi,\eta)}(x,y)= \left\{ \begin{array}{ll} e^{-x} & \textrm{when $x>0, y\in (0,1)$}\\ 0 & \textrm{...
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1answer
40 views

Have I shown $\operatorname{Var} X < \infty \iff \mathbb E[X^2] < \infty$?

Show that $\operatorname{Var}{X} < \infty \iff \mathbb E[X^2] < \infty$ I am attempting to use the above without the fact that $\mathbb E[(X-\mathbb E[X])^2]=\mathbb E[X^2]-\mathbb E[X]^2$ if $\...
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41 views

Compute $\ Var(X+Y+Z) $ where $\ X,Y,Z \sim Binomial $

Suppose I throw 3 fair dice 30 times. Let, X = no' of throws in which we don't get 4 Y = no' of throws in which we get 4 in only one die (out of 3) Z = no' of throws in which we get 4 in exactly ...
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27 views

Evaluating variance of scale parameter estimators

Let $\{Y_{i}\}_{i=1}^{n}$ be a random sample of the random variable $Y_{i}\sim \mathcal{N}(0,\sigma^{2})$, we define the following estimators for $\sigma^{2}$ $U=\frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-...
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1answer
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How to use $\mathbb E[(X-\mathbb E[X])^2]=\mathbb E[X^2]-\mathbb E[X]^2$ on $\operatorname{Var}(X)=0$

In our lecture, we have proven that $\mathbb E[(X-\mathbb E[X])^2]=\mathbb E[X^2]-\mathbb E[X]^2$ (*) then we go on say if $\operatorname{Var}(X)=0$ it follows using (*) that: $\mathbb P((X-\...
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Connection between $\operatorname{Var}(M^n v)$ and largest eigenvalue of $M$

In a proof I am trying to understand, the following is stated: $ M$ is a non-random matrix with eigenvalues $\lambda_i$, $v$ is a random vector, $n$ is a scalar, $\operatorname{Var}(M^n v) \ge \max(|...
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21 views

Scale the Variance of a Distribution

I have a distribution $\chi_{\sigma^2}$ of form, (see threebears, pqc candidate) $-1$, $\qquad$ with $P(X=-1)= \frac{\sigma^2}{2}$ $0$, $\qquad$ with $P(X=0)= 1-\sigma^2$ $1$, $\qquad$ with $P(X=...
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1answer
55 views

Calculating variance of given formula [closed]

Let $D_i$ be indicators, $i\in\{1, 2 ... k\}$. I am interested in calculating variance of $$Y = \frac {\sum_{i=1}^k x_iD_i}{\sum_{i=1}^k n_iD_i}$$ where $x_i$ and $n_i$ are given real numbers, $i \...
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$E(X(t)X(t))=\sigma^2\delta(\tau)$ or $E(X(t)X(t))=\sigma^2$ [duplicate]

Let's say we have a white noise process $x(t)$ such that: $E(X(t)X(t+\tau))=N\delta(\tau)$ $E(X(t))=0$ In particular, with $\tau=0$, $E(X(t)X(t))=E(X^2(t))$ is infinite. Now, I want $X(t)$ at each ...
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1answer
101 views

Variance of $XY(1-Y)$ in terms of the means and variances $X$ and $Y$

Consider two independent random variables X and Y. X has some distribution with mean $\mu_X$ and variance $\sigma^2_X$. Y has some distribution with mean $\mu_Y$ and variance $\sigma^2_Y$. I want to ...
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12 views

The maximum expected deviation from the sample average matrix?

I have reached to $\mathbb{E}[\|x_tx_t^T - G_t\|^2_F]$, and I need an upperbound for it in terms of probabilistic characteristics of $x_t$ where: $x_t$ is a random vector in $\mathbb{R}^n$ drawn from ...
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24 views

dividing by variance to avoid trivial solutions in multidimensional Newton's method

I'm using Newton's method as implemented by GSL (GNU Scientific Library) to try to find a cycle of a multi-variate function $F$ (reaction-diffusion simulation over two-channel 2D images collapsed to ...
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21 views

Variance of ratio of mean value of functions of a random variable

The problem: I have a random process in which the outcomes of the real valued random variable takes the independent values $x_1, x_2, ...,x_n$. Then I defined $Q$ as $$ Q = \frac{n^{-1}\sum^n f(x_i)...
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Finding the variance on the number of steps required to reach an absorbing state in a MC using the fundamental matrix

I'm trying to implement a Markov Chain class in Python to test some theoretical results with actual iterations over the matrix itself. What I've implemented up to now seems to work and output the ...
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1answer
14 views

Can off-diagonals be nonzero for covariance matrix after PCA?

I have some data for which I found the covariance matrix for: $$\Sigma = \begin{bmatrix}3.33 & −1.00 & 3.33 & 33.00 \\ −1.00 & 1.58 & −1.92 & −13.92 \\ 3.33 & −1.92 & ...
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1answer
51 views

$\mathbb{E}X^2 = \mathbb{E}Y^2$ and $\mathbb{E}(Y|\mathcal{A}) = X$ P-a.s. $\Rightarrow X=Y$ P.-a.s.

I have a question and hope you can help me. The problem is about stochastic variables $X,Y$, which are square integrable, independent and identically distributed on $(\Omega, \mathcal{S}, P)$. ...
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17 views

Calculate the variance of $f_X(x)$

We have $$f\left(x\right)=0.8\left(\frac{1}{5\sqrt{2\pi}}e^{-\frac{1}{50}\left(x-50\right)^2}\right)+0.2\left(\frac{1}{8\sqrt{2\pi}}e^{-\frac{1}{128}\left(x-60\right)^2}\right)$$ This can be written ...
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20 views

Find combination given expectation and variance

I have found the expectation of X to be 4 and the Variance to be 3. For Y the expectation is 2 and variance is 2. Is it possible to find a combination of X and Y which satisfies the expectation to be ...
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21 views

Let $s^2=\frac{1}{n}\sum_1^n( X_i-\bar{X})^2$ and $\widetilde{X}$ denote the sample median. Is $s^2\leq \widetilde{X}(1-\widetilde{X})$ true

$X_i\in[0,1]$ Let $s^2=\frac{1}{n}\sum_1^n( X_i-\bar{X})^2$ and $\widetilde{X}$ denote the sample median. Is the following true? $$s^2\leq \widetilde{X}(1-\widetilde{X})$$ I couldn't find any ...
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1answer
41 views

SOA Practice Exam: How am I to understand P(Z=z)?

Let $X$ denote the loss amount sustained by an insurance company’s policyholder in an auto collision. Let $Z$ denote the portion of $X$ that the insurance company will have to pay. An actuary ...
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1answer
20 views

$Var[W]$ when $W = X - 2Y$

I am trying to solve $Var[W]$ when $W = X - 2Y$ for independent discrete RVs $X$ & $Y$: $$E[X] = 9, Var[X] = 4; E[Y] = 2, Var[Y] = 1.$$ I understand $$ Var[W] = Var[X] + Var[2Y]$$ $$= ...
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Variance of the stochastic process

I am asked to find $Var(aW_t+b\int_0^tf(s)W_sds)$. This is what I did until now: \begin{align*} &\mathbb{E}\left(a^2W_t^2+2abW_t\int_0^tf(s)W_sds+b^2\left[\int_0^tf(s)W_s \, ds\right]^2 \right) \\...
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1answer
31 views

How can I find the expectation and variance of $Z=\max\{X,Y\}$ where $X$ and $Y$ are defined through joint probability distribution?

Random variables $X$ and $Y$ and have the joint distribution below, and $Z=\max\{X,Y\}$ $$ \begin{array}{c|lcr} \text{X\Y} & \text{1} & \text{2} & \text{3} \\ \hline 1 & 0.12 & 0....
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Showing Rejection Region Equality with Fisher Distribution

I'll preface this with saying that this question is a homework question, but not one that is graded or turned in in any form. Specifically it is from the text Mathematical Statistics with Application ...
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21 views

Coin toss- expected value.

A coin had tossed three times. Let: $X$-number of tails $Y$-number of heads Find the expected value and variance $Z=XY$ My solution: $E(Z)=E(XY)= 2 \cdot 1/4 + 3 \cdot 1/4=6/4$ Because, I know that $...
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1answer
23 views

Expected number of good presents

Given $b$ boys and $g$ girls. Children give presents to each other. They know who gives a present whom from random permutation of $1,2,\dots b+g$. If child gives present to child with same gender ...
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2answers
31 views

Covariance and Law of Large numbers

Say I am taking the average value of the product of two dependent random variables $X$ and $Y$ sampled an infinite amont of times. That is I am computing $\lim_{n \rightarrow \infty} E \left[ \sum_{i=...
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1answer
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Bivariate normal: Expected value and variance

I have this simple exercise to do and I'm new to this topic. Seeing the slides of my professor, I would solve the problem in this way: $E(Y)=c_1 μ_1+c_2 μ_2$ $E(Y)=-54.2424$ $Var(Y)=σ_{22}$ $Var(...
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Find the expected value and standard deviation of the returns for the following portfolio

Suppose you want a portfolio composed of AT&T, Cigna, Disney, and Ford. Find the expected value and standard deviation of the returns for the following portfolio I know that I have to use ...
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1answer
23 views

Expectation and variance of travel time with several options for the transportation

A person is traveling between two places, and has 3 options for transportation. The jth option would take an average of µj hours, with a standard deviation of $\sigma_j$ hours. The person randomly ...
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1answer
26 views

what premium should the company charge each policy holder to assure that the premium income will cover the cost of the claims?

A car insurance company has $2,500$ policy holders. The expected claim paid to a policy holder during a year is $1,000$ with a standard deviation of $900$. What premium should the company charge ...
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1answer
31 views

Calculating the variance of a random variable

I have stumbled upon this question from my text book and I have been finding difficulty in understanding and solving it. If X and Y are independent random variables with variances $σ_X^2 = 5$ and $...
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1answer
44 views

Conditional mean and variance of $X$ given that $Y=6$ for $X$ normal $N(0,1)$ and $Y$ conditionally on $X=x$ normal $N(x,1)$

Assume that $X\sim N(0,1)$ and that, given $X=x$, the conditional distribution of $Y\mid X=x$ is $N(x,1)$. Find the conditional mean and variance of $X\mid Y=6$. I proceeded as follows: $$f_{X|Y}(X|...
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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 ...
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Why are the negative estimates of variance components replaced by zeros in ANOVA?

I have a question regarding a well sought technique in ANOVA that the negative estimates of variance components are often replaced by zeros.Yes, I know that it is sensible but why is it made to $0$?
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1answer
24 views

Is $E[g(x_i)g(x_j)]=E[g(x_i)]\,E[g(x_j)]$, for $x_i$ multinomial?

Consider the multinomial distribution (Wikipedia): and let $g:\mathbb{R}\longrightarrow \mathbb{R}^+$ a smooth function. I would like to know if one can show the identity $$E[g(x_i)g(x_j)]=E[g(x_i)]\...
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2answers
21 views

Expected value and variance of a transformed random variable

X is a binomial random variable with $n = 10000$ and $p = 60\%$ (hence $E(X) = 10000 * 60 = 6000$). Now Y is a transformed random variable with $Y = 100000 - 7X$. I have to calculate the expected ...
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1answer
30 views

On the variance in the Bernoulli scheme

I'm trying to solve the following problem. There are $755$ cards, each of which has a number from $1$ to $755$ (each number occurs exactly once). Of these cards, $20$ pieces are randomly selected ...
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1answer
42 views

Expectation and variance of number of movie tickets

Let $N\sim\mathrm{Pois}(\lambda_1)$ be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is $T\sim\mathrm{Pois}(\lambda_2)$, independently. ...
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19 views

Variance definition (basics)

\begin{align}V(X)&=\sum_x(x-E(X))^2P(x)\tag{1}\\V(X)&=E([X-E(X)]^2)\tag{2}\\E(X)&=\sum_xxP(X=x)\tag{3}\end{align} Can someone please explain me how I derive equation (2) from equation (1)?...
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1answer
36 views

Show that variance function is non-negative over all arguments

The variance function of the classical occupancy distribution is given in various references$^\dagger$ as: $$V(n,m) = m \Bigg[ \Big( 1 - \frac{1}{m} \Big)^n + (m-1) \Big( 1-\frac{2}{m} \Big)^n - m \...
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1answer
49 views

Marginal and Posterior Distributions

An election is being held. There are two candidates, A and B, and there are n voters. The probability of voting for Candidate A varies by city. There are m cities, labeled 1, 2, . . . , m. The jth ...