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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Having trouble with given task

The random size is distributed according to the normal law, which is an average of 15. Calculate this random size dispersion if it is known that the probability of gaining values ​​from the range [15; ...
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1answer
23 views

Prove that $\hat{\sigma^2}=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is not an efficient estimator.

I am asked to show that the unbiased estimator $\hat{\sigma^2}=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is not efficient. So far I was able to show that the Rao-Cramer Lower Bound is $\frac{2\sigma^...
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1answer
22 views

Variance of Binomial Distribution - formula

This might be a really stupid question... but still here we go: For the Variance we have two formulas: (i)$$ \sigma^2:=\sum_{k=0}^n (k-\mu)^2p = \sum_{k=0}^n (k^2-2npk+n^2p^2)p= (\sum_{k=0}^n k^2- \...
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9 views

How to calculate the variance

So i have this question and I wanted to be sure that my calculations were correct. The standard deviations of the market returns is $0.2(20%)$ and the covariance between the return on the market ...
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1answer
32 views

How to calculate the variance of a portfolio?

So I have this question : Florida Company (FC) and Minnesota Company (MC) are both service companies. Their stock returns for the past three years were as follows: FC: -5 percent, 15 percent, 20 ...
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0answers
7 views

Variance of two random variables that are defined by independent variables

Let $$Λ_1 = \frac{1}{4} Λ^{(2)} + \frac{3}{4} Λ^{(4)},$$ $Λ^{(2)}, Λ^{(4)}$ are independent random variables. $\mathbb EΛ^{(s)}=\frac{s+5}{s^2}$ and $\mathbb DΛ^{(s)}=\frac{(s+5)^2}{s^2}.$ From the ...
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13 views

transform variance to constant

Suppose I have a sequence of data $\{x_t\}_1^N$ whose variance is a function of time, $\sigma^2(t) = \sigma_0^2 *t$, where $\sigma_0^2$ is a constant. How can I transform the variance of the entire ...
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6 views

Proof of Fermi Estimation Variance

On the wikipedia page https://en.wikipedia.org/wiki/Fermi_problem, it claims that "In continuous terms, if one makes a Fermi estimate of n steps, with standard deviation σ units on the log scale from ...
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2answers
27 views

Barttletts test and F test with 3 variables [on hold]

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
16 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
34 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
22 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 ...
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3answers
39 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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0answers
14 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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11 views

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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3answers
50 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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0answers
21 views

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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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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17 views

$Var(\frac{1}{nh^d}\sum_{i=1}^n Z_i)=\frac{1}{n}E\bigg[\big(\sum_{i=1}^n Z_i-E(Z_i)\big)^2\bigg]$

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
14 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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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
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<\...
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1answer
18 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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21 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
58 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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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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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
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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
38 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
14 views

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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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
43 views

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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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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0answers
16 views

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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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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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
29 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
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) = \...