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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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Barttletts test and F test with 3 variables

I linked a picture with the assignment. I know how to do F-tests with 2 variances and means, but with 3, I am out of luck. I could do it if I had a dataset in R. but not manual in R or by hand, as it ...
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0answers
13 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 ...
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1answer
33 views

Given $E[X] , \operatorname{Var}(X)$ and $Y\mid X \sim U(X,1)$, find $E[Y]$ and $\operatorname{Var}(Y)$

For $X, Y$ random variables, given $E[X] = \mu$ ; $\operatorname{Var}(X) = \sigma^2$; $Y\mid X \sim \text{Unif}(X,1)$: Find $E[Y]$ and $\operatorname{Var}(Y)$. (1) To find E[Y], I used the law of ...
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0answers
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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 ...
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2answers
4k views

Conceptual Understanding of Rate / Volume Analysis for Balance Sheet changes

Not sure if this is the right place to ask this but I searched and didn't find this question already asked. I am having a lot of trouble conceptually understanding the formulas behind a rate / volume ...
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3answers
36 views

Why are there two formulas for the sample variance?

I'm using an introductory statistics textbook and it mentioned these two formulas for the sample variance: $s^2 = \frac{\sum(x - \bar{x})^2}{n - 1}$ and $s^2 = \frac{\sum{}x^2 - \frac{(\sum{}x)^...
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1answer
25 views

Probability $P(\mu - d\sigma < X < \mu +d\sigma) \ge 1-\frac 1 d^2$

Show, that $P(\mu - d\sigma < X < \mu +d\sigma) \ge 1-\frac 1 d^2$ , when $\mu < \infty$ and $\sigma^2>0$ and $ d>1$ My Idea: $P(\mu - d\sigma < X < \mu +d\sigma)$ = $1-P(X \ge \...
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2answers
820 views

Let $T_1$ be an unbiased estimator of $\theta$. Show that $T_1^2$ is a biased estimator of $\theta^2$.

Let $T$ be an unbiased estimator of $\theta$. Show that $T^2$ is a biased estimator of $\theta^2$. We have $\operatorname{E}(T)=\theta$ by assumption. Then $\operatorname{E}(T^2)=\operatorname{Var}(T)...
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0answers
13 views

Deriving bootstrap variance of absolute central moments

My statistics text introduces the bootstrap as follows: Draw $X_1^{*},\ldots,X_n^{*} \sim \hat{F}_n$. Compute $T_n^{*} = g(X_1^{*},\ldots,X_n^{*})$ where $g$ is any function you want to ...
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Signal-to-noise ratio

Consider the linear structural equation model with known $\beta$ as $$ SEM \quad X_k = \sum_{j=1}^{p}\beta_{jk}X_j + \epsilon_k$$ where $X_k$ is a random variable. I construct a data matrix $D_{m \...
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1answer
1k views

Minimum mean squared error of an estimator of the variance of the normal distribution

I am trying to find the estimator of the variance $\sigma^2$ of a normal distribution with the minimum mean square error. From reading up, I know the unbiased estimator of the variance of a Guassian ...
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3answers
49 views

Probability- 6 digit number that is built from the numbers 2,5,6,9

A random 6 digit number is picked that is built only from the numbers $2, 5, 6, 9.$ What is the probability that the number can be divided by $3$? What is the probability that the number can be ...
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1answer
21 views

Measuring the uniformity or closeness of a set of given values

Say we have a sample space of size $N$ . What's a good way to measure how close the values are to one another? In other words, to what degree are the values 'equal' to one another? I thought of a ...
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0answers
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Find covariance matrix of $\frac{f(x + y) }{x + y}$ function

The conditions are the same, but my task is to find covariance matrix. I only noticed that density function is symmetric so expected values, variance are also the same. But I don't know how to find ...
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3answers
34 views

Calculating Variance(X + Y + 1)

Not Duplicate. I know a question with similar data has been used here, but I am looking for something else. Two tire-quality experts examine stacks of tires and assign a quality rating to each tire ...
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0answers
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$Var(\frac{1}{nh^d}\sum_{i=1}^n Z_i)=\frac{1}{n}E\bigg[\big(\sum_{i=1}^n Z_i-E(Z_i)\big)^2\bigg]$

Let $\{Y_i,X_i\} \in \mathbb{R}\times\mathbb{R}^d$ be a strictly stationary sequence of random vectors and consider $\hat{\Psi}(x)=\frac{1}{nh^d}\sum_{i=1}^nY_iK(\frac{x-X_i}{h})$, where $h=o(1)$ is ...
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1answer
11 views

Sequence of independent random variables with same expected value such that the weak law doesn't hold

I'm looking for a specific counterexample for the weak law of large numbers. That is, I want a sequence of random variables with same finite expected value $\mu$. These random variables must each have ...
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1answer
739 views

Standard deviation transformation sd(X-Y)

just need some help in this statistics question regarding the transformation of standard deviations. This is the question: Let R be the visual acuity readings for the right eye of a randomly ...
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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
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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
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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<\...
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1answer
17 views

What is the standard deviation of the frequency table below?

Hello, I have some issues with finding the standard deviation of this question. I found the mean as 337.5 km, and I considered that standard deviation is the root of the variance. I first squared the ...
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1answer
26 views

Estimate variance in a nested study design

I have a "mother" bacteria. I cloned it $R$ times and measured a variable $P$ on these $R$ clones and recorded the mean $\bar p$ and the variance $V$ among these clones. Then, I made $M$ mutants ($M$ ...
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1answer
43 views

Proving $V(X) = E(V(X|Y)) + V(E(X|Y))$ using the pythagorean theorem

I know the textbook proof of $$V(X) = E(V(X|Y)) + V(E(X|Y))$$ but I'm interested in understanding the weird proof/analogy with the pythagorean theorem my professor gave in class. With $X, Y$ random ...
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0answers
20 views

Variance of random variable (Langevin Equation)

I am trying to solve the following problem and I am slightly confused/stuck. The Langevin equation reads: $$\gamma \frac{d x(t)}{d t} = -\frac{\partial V(x)}{\partial x} + \zeta(t) \tag{Eq. 1}$$ ...
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1answer
56 views

Why are there two formulas for variance of random variables?

I'm using an introductory statistics textbook and it mentioned this: Definition: If $X$ is a random variable with mean $E(X) = \mu$, then the variance of $X$ is defined by $Var(X) = E((X−\mu)^2)$....
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1answer
14 views

Variance of an estimator

I was wondering how to compute the variance of the following estimator for the mean. Where $\{Y_1,Y_2,Y_3\}$ is a sample from an exponential distribution : $$\hat{\theta} = \frac{Y_1 + Y_2}{4} + \...
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How to determine the variance of this Bayesian linear regression model

I am struggling with the following problem. I have a Linear regression model which uses bayesian statistics: $X_7 = X\beta + \epsilon$. Here $X = (X_0,X_1,...,X_6)$ is a data frame containing 7 ...
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3answers
44 views

How to calculate covariance with an minus like V(X-Y)?

my task is this: Be $ X $ and $ Z $ independent with the same distribution and $ Y :=X-Z . $ Calculate $ \operatorname{cov}(X, Y) $ and $ \operatorname{corr}(X, Y) . $ My Problem is the minus in $...
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1answer
37 views

How to calculate E[Xi Xj]?

This question is from an example in the book of Bertsekas. (p240 of 1st edition). I would like to know why $$E[X_{i} X_{j}] = P(X_{i} = 1\text{ and }X_{j}=1)$$ and $$E[X_{i}] = P(X_{i}=1)$$. please ...
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1answer
16 views

Law of total variance and covariance given X and Y are normal

I have a problem which asks me to find $\Bbb E[Y]$ and $Var(Y)$ given that $Y\text{~}Normal(x,1)$ conditional on $X=x$. $X$ is standard normal. So I have worked out that $\Bbb E[Y]=0$ using the law of ...
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1answer
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Variance of inverse gamma distribution

Given a random variable $X$ which is distributed gamma with shape $\alpha$ and rate $\lambda$, for which the variance is known, how does one calculate $\text{Var}(\frac{1}{X})$? I am hoping not to ...
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1answer
23 views

DEMONSTRATION FINITE-SAMPLE PROPERTIES OF LEAST SQUARES $\frac{(N-k)S^2}{\sigma^2}\sim\chi^2[n-K]$

Im a Student of Economics, and I have a concern. In the solution of $\frac{(n-K)S^2}{\sigma^2}\sim\chi^2[n-K]$ How can I show that if the matrix is ​​symmetric and idempotent between $(I-H)=|| (I-...
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1answer
19 views

Difference of normal random variables

I have two random variables $$X_{s+t} \sim N(0, s+t)$$ $$X_s \sim N(0, s)$$ where $s \leq t$. How do I show that... $$X_{s+t} - X_s \sim N\left(0,s + t + s -2\sqrt{s(s+t)} \right)??$$ I understand ...
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2answers
41 views

How to calculate the bias of the estimator for variance?

Question: For observations $x_1$, $x_2$, . . . , $x_n$ with sample average $\bar{x}$, we can use an estimator for the population variance: $\hat\sigma^2$ = $\frac1n\cdot $ $\sum\limits_{i=1}^n (x_i - ...
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2answers
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How to use the law of total variance

I know that the law of total variance states $$Var(X)=\Bbb E[Var(X|Y)]+Var(\Bbb E[X|Y])$$ But how does one treat $Var(X|Y)$ and $\Bbb E[X|Y]$ as random variables? For example, say we know that $$\Bbb ...
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1answer
52 views

$Y = \frac{X_1 X_2}{X_3}$ where $X_i$ is a uniform random variable

$Y = \frac{X_1 X_2}{X_3}$ where $X_i\sim U(0,1)$ and $X_1,X_2,X_3$ are i.i.d I need to calculate $Var(Y)$ and $Var[Y|X_3=1.7]$ I know that for each $X_i$, $E[X_i]=\frac{1}{2}$ $Var[X_i]=\frac{1}{...
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2answers
1k views

Expected value and variance of a piecewise function with the integral

I have the following stepwise function: f(x) = 1/3 for -1 <= x < 0, 2/3 for 0 <=x<=1, 0 otherwise I wonder how I can derive E(X) and Var(X) of a stepwise uniform function using the ...
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0answers
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Proving convergence of expectation and variance given Rényi's $\alpha$-divergence tends to 0

I denote $p, q$ as density function of $P, Q$. Given $Y, X$ are random variables and \begin{align} \int q(x)\mathbb{D}_{\alpha}[p(Y\mid X=x)\,||\,p(Y\mid x^\star)] \,dx \rightarrow 0 \end{align} ...
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0answers
25 views

Proving expectation and variance of a function of a random variable tends to a fix point

Given $f:\mathcal{X} \rightarrow \mathbb{R}$ is a continuous function and $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ ($x^\star$ is a fix number), $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$. How can ...
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2answers
24 views

Can Conditional Expected Value be negative in normal distribution?

So, the problem gives me this facts (for a Normal bivariate distribution X,Y) $$Var(Y|X=x) = 5$$ $$E(Y|X=x) = 2 + x$$ It asks me to find $$E[Y^2|X=7]$$ I tried this: using the conditional variance ...
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2answers
704 views

Decomposition of variance

Suppose X is a continuous random variable that can take any value between plus and minus infinity. Furthermore, suppose A is a random variable capturing those events where X is below 0, and B is a ...
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1answer
17 views

For a sequence of experiments where each $X$ is the number of trials until success with varying $p$, is each $X$ independent?

Assume that, every time you buy a box of Wheaties, you receive a picture of one of the $n$ baseball player. Let $X_k$ be the number of additional boxes you have to buy, after you have obtained $k-1$ ...
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1answer
16 views

Finding the probability of success that maximizes the variance of independent trials

A professor wishes to make up a true-false exam with n questions. She assumes that she can design the problems in such a way that a student will answer the jth problem correctly with probability $...
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1answer
12 views

Find mean and variance from mgf where t is denominator

For continuous random variable X, pdf: $f_{X}(x)=2(1-x), x\in[0,1]$ mgf: $M_{X}(t)=\frac{2(e^t-t-1)}{t^2}$ Problem is to find mean and variance from mgf, I tried using $\frac{d}{dt}M_{X}(0)$ and $\...
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1answer
28 views

Rao-Blackwell and Cramer-Rao LB comparison

Let $X_1, X_2, \dots, X_n$ be a random sample following the Geometric distribution. $$ \prod\limits_{i=1}^{n} f(x_i|p) = (1-p)^{\sum\limits_{i=1}^n x_i-n}p^n $$ Since the pmf of the Geometric ...
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2answers
34 views

Difficult demonstration - How to show that $H_n$ is normal distributed $N(\xi,\sigma^2)$ starting from its moments $ξ$ and $σ$?

I was thinking that if the function $H_n$ of cumulative distribution converges to a distribution $H$, then $\epsilon_n$ should converge to $\epsilon$ what could be expressed as follows: If $H_n$ is ...
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1answer
778 views

Intuition behind the critical difference of ANOVA

I am studying Analysis of Variance. Suppose, we have done ANOVA and found out that null hypothesis is rejected. This means there is a significant difference between at least one pair of the ...
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1answer
41 views

Law of total covariance inequality

The law of total variance: $$ \text{Var}(X) = \mathbb{E}(\text{Var}(X\mid Y)) + \text{Var}(\mathbb{E}(X\mid Y)).$$ There is also something called the law of total covariance: $$ \text{Cov}(X,Y) = \...
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0answers
18 views

Variance of linear combination

This is a follow up question to this. Let $(X_1,\ldots, X_n)$ be non-independent random variables such that $$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$ where $\mathcal{N}(0,1)...