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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.

13
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239 views

Trimmed mean: Take $n$ i.i.d. Gaussians and remove largest $m$ and smallest $m$ points. What is the variance of the mean of the remaining points?

Let $n,m\in\mathbb{Z}$ with $0 \le 2m < n$. Let $X_1, \cdots, X_n$ be i.i.d. standard Gaussians and let $X_{(1)} \le X_{(2)} \le \cdots \le X_{(n)}$ denote their order statistics (i.e., $\{X_1, X_2,...
5
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129 views

Supermartingale variance bound?

Suppose I have a supermartingale $$ \mathbb{E}[X_{n+1} \mid X_n, \dots , X_2, X_1] \leq X_n $$ There are 2 other conditions: bounded difference: $|X_{n+1} - X_n| \leq A$ with probability $1$ (...
4
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0answers
104 views

The variance of a covariance

The expectation over $x$ of a covariance between variables $A$ and $B$ (where the distribution of $B$ varies according to $x$) is equal to the covariance of $A$ with the expectation of $B$ over $x$: $...
4
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0answers
55 views

Seemingly Identical Random Variables with Different Variances

In my probability class, we did the following problem regarding expected values/variance: Consider an experiment where you roll a fair, 6-sided dice until you see a 6. Let $T$ be a random variable ...
3
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39 views

Help me find my mistake for this variance

Suppose X is an observation from a distribution with probability mass function $$X\sim f(x)=\left(\frac{\theta}{2}\right)^{\left | x \right |}(1-\theta)^{1-\left | x \right |} 1_{A}(x)$$ $$0<\...
3
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0answers
53 views

Sum of Discrete Uniform Variables times extraction index

Let $X_i$ be the $i-th$ extraction from an urn with $N$ balls numbered from $1$ to $N$. Let's make $N$ extractions without replacement, so that that the urn is left without balls. Let $Y_i = X_i \...
3
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246 views

Mean Value and Variance of a Birth and Death Process

Let $\{X(t)\}_{t>0}$ on $\{0,1,2,3\}$ a birth and death process, with $\lambda(s)=(3-s)^2$ and $\mu(s)=s^2+s$. Assume $P(X(0)=3)=1$ and determine: (a)$E[X(t)]$; (b)$Var[X(t)]$. I don't know how ...
3
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4k views

Derivation of Variance of Discrete Uniform Distribution over custom interval

I'm trying to prove that the variance of a discrete uniform distribution is equal to $\cfrac{(b-a+1)^2-1}{12}$. I've looked at other proofs, and it makes sense to me that in the case where the ...
3
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75 views

Expectation and variance (unknown problem type)

I'm having trouble identifying what type of problem the following question is: A student is trying a new study strategy for the final exam. There are four topics to study for the exam and each day ...
2
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51 views

Why can we use a consistent variance matrix estimator when finding the asymptotic distribution

So I am pretty sure that in a one-dimensional case, we would just say $x \overset{d}{\to} N(0,\sigma^2)$ and $s^2 \overset{p}{\to} \sigma^2$ so $\frac{s}{\sigma} \frac{x}{s} \overset{d}{\to} N(0,1)$ ...
2
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37 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) + \...
2
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24 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 ...
2
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44 views

Brownian Motion and Variance

If $(W_t) _t$ is a Brownian motion regarding to a filtration $(F_t) _t$ and the process $Z(t)$ is defined by $$ Z_t= \int_0^t W_s ds$$ What is $\operatorname{Var}(Z_t)$?
2
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55 views

The joint density of X and Y is given by…

The joint density of $X$ and $Y$ is given by $$f_{X,Y}(x,y)= \left\{\begin{matrix}(x+y), \mbox{ } 0\leq x \leq1 , 0\leq y \leq1 \\ 0, \mbox{ otherwise}\end{matrix}\right.$$ a) Evaluate $f_{X}(X)$ ...
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25 views

Variance estimation question and unbiasedness.

Hello I have a problem with variance estimation for RQMC. I have $\{x_1,...,x_n\}$ Sobol points. A randomised set $\tilde{\mathfrak{X}}$ is generated by the random variable $\epsilon$. For a ...
2
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0answers
44 views

Hint for computing the mean and variance of $\hat{\alpha} = - \frac{n}{\log \prod_{i=1}^n X_i }$

In my statistical class, we are learning MLE. Last week we end up computing the MLE of parameter $\alpha$ for a random sample $X_1, X_2,...,X_n$ with $\text{Beta}(\alpha,1)$ distribution with support $...
2
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0answers
33 views

Find the variance of Y

In an old probability test: $$ P(X_i=1)=n^{-\frac{1}{2}} $$ $$ P(X_i=0)=1-n^{-\frac{1}{2}} $$ $$ S_{i,j,k} = 1 \text{ if }X_i=X_j=X_k=1 \text{ (and 0 otherwise)}$$ $$ Y = \text{Number of }S_{i,j,k}\...
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375 views

third central moment negative binomial distribution

Let $X$ be negative binomial distributed with parameter $p$ and $r$. Show that: $1$) $E$($X$) = $r$ $\cdot$ $\frac{q}{p}$ $2$) $Var$($X$) = $r$ $\cdot$ $\frac{q}{q^2}$ $3$) Calculate the third ...
2
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287 views

Variance for the function $g(x)=\frac{1}{(1.1-x)^2}$ where x is has uniform distribution $U(0,1)$

Calculating the mean of the random variable $g(x)=\frac{1}{(1.1-x)^2}$ where x is has uniform distribution $U(0,1)$: $E[g(x)]=\int\limits_{0}^{1}\frac{1}{(1.1-x)^2}dx=\frac{1}{0.1}-\frac{1}{1.1}=9.09$...
2
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0answers
227 views

Expected Volume of Convex Hull of Spherical Normal Distribution

I would like some input on if my understanding and solution to the following problem is correct: Given a spherical normal distribution $N(\mu, \Sigma)$ of dimension $d$ with: $$\Sigma = \left[ ...
2
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0answers
54 views

Variance calculations: sum of powers of matrices

I'm facing the following problem: let $V = \sum_{t=0}^{\infty} P^{t}$ where P is a $n \times n$ matrix. I want to calculate $Var(V)$, variance is calculated coefficient-wise. I know the variance of ...
2
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0answers
202 views

Markov Chain: Stationary distribution variance

Suppose to consider a generic Markov Chain, suppose $P$ to be the unknown transition matrix and $\pi$ its stationary distribution. Assume $Var[P]$ to be known (estimated from some data). (Variance ...
2
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0answers
102 views

Is weighting by relative variance a meaningful way to weight variables?

For a school project, I am trying to determine similarity between participants based on different aspects of their athletic mechanics. I have the same number of observations per participant, and 8 ...
2
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0answers
212 views

Mean and variance of a Random variable

I am trying problem no. 2 from Purdue HW here . Balls are drawn from an urn containing $w$ white balls and $b$ black balls until a white ball appears. Find the mean value $m$ and variance $σ^2$ of ...
2
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0answers
74 views

Simplify variance expression, taking into account covariances:$\operatorname{var}(Y_i-\bar Y-\hat {\beta}_1(x_i-\bar x))$

Find $\operatorname{var}(Y_i-\bar Y-\hat {\beta}_1(x_i-\bar x))$ where $\hat {\beta}_1=S_{xy}/S_{xx}$ is the least square estimator and $Y_i$ a random variable. I know that I can't simply split the ...
2
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0answers
677 views

Variance of a sum of identically distributed random variables that are not independent

I am "new" to probability/statistics and I was hoping someone could verify that this is correct. Let $Y_1,\ldots,Y_n$ be random variables that follow a common distribution with mean $\mu$ and variance ...
2
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0answers
386 views

Stationary Gaussian process whose correlation parameter approaches zero.

Consider a mean-zero ($\mu = 0$), unit-variance ($\sigma^2$) Gaussian random process $X(t)$. This process is strictly stationary (the joint-probability distribution does not vary with $t$). The ...
2
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0answers
87 views

Means and Covariances of powers of a normal distribution

Let $X$ be a normally distributed random variable, with mean $\mu$ and variance $\sigma^2$. Consider a random vector $$V = \left[ X^n, X^{n-1}, \dots, X^2, X, 1 \right]^T $$ What is the expected ...
2
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0answers
56 views

Show that $E(\mathrm{var}(Y|X)) \leq (1 - \mathrm{corr}(X,Y)^2) \mathrm{var}(Y)$

Expectation of conditional variance (Exercise 4.6.7 from Grimmett and Stirzaker): Let $X$ and $Y$ be random variables with correlation $\rho$. Show that $E(\mathrm{var}(Y|X)) \leq (1 - \rho^2) \mathrm{...
2
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0answers
56 views

Under which conditions on $\sigma_1, \sigma_2$ and $\rho_{12}$ the minimum variance portfolio involves no short selling?

If $\rho_{12} \lt 1$ or $\sigma_1 \ne \sigma_2$ then $\sigma_{V}^2$ representing the variance of the portfolio with weights $(w_1, w_2)=(s, 1-s)$ as a function of $s$ attains its minimum value at $$...
2
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0answers
80 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal N(\...
2
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0answers
132 views

Convergence of Sample Mean Via WLLN

I am trying to show that the sample variance converges to the population variance in using the Weak Law of Large Numbers $$\begin{align} \\ \Rightarrow S_n= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2 &...
1
vote
0answers
11 views

Variance service times M/G/c queue

I am wondering about the influence of the variance of the service times in an M/G/c queue on the probability that a customer has to wait. Intuitively, I would say that smaller variance implies smaller ...
1
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0answers
18 views

Calculate p-value between two lists of floats of unequal size

I would like to calculate the degree of variance between to lists of floats of unequal size expressed in a p-value. I tried a two-sided t-test as in the example below using python. But answers that ...
1
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0answers
50 views

Which is faster: a bank with five lines of ten or one line of fifty?

I'm working on a probability question with mean and variance. Let's say that I have two banks. They are identical in every way, except that bank A has five lines with ten people and bank B has ...
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0answers
41 views

Value At Risk Elliptical Distributions

I thought I understood elliptical distributions, but then I staggered over the following problem: Let d financial returns be modeled as the components $X_1,...X_d$ of a d-dimensional random vector $X$...
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0answers
25 views

Sum of exponential series of equal mean and variance

Assuming $A$ and $B$ are two non-negative real-valued random variables such that $\mathrm{E}(A)=\mathrm{E}(B)$ (equal means) $\mathrm{Var}(A)=\mathrm{Var}(B)<\epsilon$ (equal small variances) is ...
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0answers
20 views

Distribution/Variance of correlated squared normal random variables

If $X_{1}, X_{2}, \ldots, X_{N}$ are identically distributed normal random variables with mean $0$ and variance $\frac{(N+3)D\sigma^{2}}{N}$, then I want to calculate the distribution, or at least the ...
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0answers
41 views

Linear regression and constant conditional variance

This is an exercise found in Mathematical Statistics with Applications, by Freund. The book defines the regression equation of $Y$ on $X$ as $$ \mu_{Y|x} = E[Y|x] = \int_{-\infty}^{\infty}yf(x|y)dy $$...
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0answers
23 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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0answers
34 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 ...
1
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0answers
74 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} -...
1
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0answers
15 views

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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0answers
95 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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0answers
35 views

central limit theorem: what is the variance?

This is a very basic question that I'm pretty sure I understand but I wanted to double check. Given a regression model of: $$y_t = \mathbf{x_{t}^{\prime}}\beta + u_t$$ We can use one of the CLT ...
1
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0answers
19 views

Throw a dice-expected value.

We throw the dice three times. Let $X_i$- number of throws, in which we get number $i$. Find expected value and variance random variable $Y=\sum_{i=1}^{6} (-1)^iX_i$. We know, that $X_i=0,1,2,3$. $E(Y)...
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0answers
16 views

Simplifying an equation with covariances of random vectors

Let $I=\begin{pmatrix}I_1\\\vdots\\ I_n \end{pmatrix}$ and $X=\begin{pmatrix}X_1\\\vdots\\ X_p \end{pmatrix}$ be two random vectors and $\Omega_I$ a random variable. I am looking for A such that: $$\...
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0answers
34 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}-...
1
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0answers
17 views

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(|...
1
vote
0answers
20 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 ...