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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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$Y = \frac{X_1 X_2}{X_3}$ 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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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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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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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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10 views

Weibull distribution -variance, expected value [on hold]

If $X \sim Exp(\lambda)$, so what is a variable distribution of $Y=X^{\frac{1}{\alpha}}$ for $\alpha >0$? Calculate the expected value, variance and intensity of failures. For which the value of $\...
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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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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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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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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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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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37 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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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)...
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Why is the standard deviation described as $\sqrt{pqn}$ sometimes and sometimes as $\sqrt{\frac{pq}{n}}$?

I assume it has something to do with whether we start with a distribution or with samples, but why is the standard deviation increasing with $n$ in one case and decreasing in the other?
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Mean and Variance in the Jacobi Stochastic Volatility model

I would like to compute $ E[X_{T}]$ and $Var[X_{T}]$ in the Jacobi model, where the Dynamics are given as $ dY_{t}=\kappa(\theta-Y_{t})dt+\sigma\sqrt{Q(Y_{t})}dW_{1t}$ $dX_{t}=(r-\delta-0.5Y_{t})dt+\...
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38 views

Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$

I am working on this problem. Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$ . So far I am thinking of using the invariant property of MLEs, so I let $$\hat{\theta} = \...
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Determining if 2 variances are equal with a rule of thumb?

So yes I am familiar with the F-test and know how to use it. Though I remember there was a quick rule of thumb of determine if 2 variances are equal. By either subtracting or dividing the 2 and it ...
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10 views

Mean and variance of conditional mean and variance

I was given two discrete r.v. $X$ and $Y$. I know how to compute $E(X|Y=y)$ and $V(X|Y=y)$ and i realize how you could treat $E(X|Y)$ and $V(X|Y)$ as random variables them self. Yet I'm ...
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Reference relating variance and covariance to elementary linear algebra for undergraduates

I'm currently teaching a non-calculus probability and statistics course for business students, and I have a student who is interested in why we use the variance instead of the Mean absolute deviation(...
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Variance and covariance inequality

Given a real-valued random variable $X$, is $$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$ true? Any pointers for how to tackle this problem would be immensely helpful.
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Does measuring the variance above the mean only give you a better indication of ceiling/potential?

I want to find NBA player's who have a better chance of scoring an extremely high number of points based on their average and variance. However, with total variance you get numbers that also fall ...
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Trouble with seemingly extremely simple statistics question involving normal distribution and expectation.

Say we have a random variable X which is distributed like such: X ~ N(1, 4). A question asks me to calculate $E(X^2)$, which I thought would be straight forward. I use the formula for Variance: $Var(...
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Calculating variance of a sum

The number of students per day has the distribution N ∼ Poisson(10). The students of CSUEB withdraw money from a cash machine according to the following probability function (X): X | 50 | 100 ...
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Show that the autocovariance function depends on $s$ and $t$ only through their difference $\left|s-t\right|$

Consider the time series $$ x_t = \beta_1 + \beta_2t + w_t, $$ where $\beta_1, \beta_2$ are known constants and $w_t$ is a white noise process with variance $\sigma^2_w$. I want to show that the ...
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40 views

Inequality for Uniform Distribution

Let $X_1,..,X_n$ be a random independent sample $X_1,…,X_n$ from a Uniform$[0,\theta] $ distribution, $\theta \in [0, \infty)$, with probability density function $f(x;\theta) = \begin{cases} 1/\theta,...
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24 views

mean for cumulative distribution function

I'm trying to find the mean (expected value) and variance for the following distribution function: $F(x)=\begin{cases} 0 & \text{for } x \lt 1\\ \frac{x^2-2x+2}{2} & \text{for } 1 \...
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Variance of Residuals in Multiple Linear Regression

If I have $n$ variables $x_1,\dots,x_d$ in my dataset, and I regress the first one against all the others i.e. $x_1=a_{12}x_2 + a_{13}x_3 + \dots + a_{1d} x_d + \epsilon_1$, how can I calculate $\...
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What is the expected value and variance of a random variable that is based on another random variable?

Im unsure if this is the correct question, sorry. Let B be the a random variable with expected value 10 and variance 4 that is defined on B >= 0. If Y = 4 + B/10, what is the expected value and ...
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23 views

Calculate expected value and variance of a t-student distribution without using density function

Let $(X_n)_n$ a suite of random variables independent and identically distributed, $X_i \sim \mathcal{N}(0,1)$ and let $Y_n:= \sum_{j=1}^n X_j ^2 \sim \chi^2_n$ a chi-square random variable with $n$ ...
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26 views

Conceptual difference: infinite mean and non-existent mean

Let $X$ be a random variable with finite variance. I know that this implies a finite mean. Do I have to prove that or does it follow from the definition of the variance? The variance is defined as $...
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Finding Variance of Piecewise Function of Two Random Variables

I have a piecewise function of two random variables: $$h(X,Y) = \left\{ \begin{array}{ll} kXY \qquad \text{if } X\geq a\\ kaY \qquad \text{ if } X < a\\ ...
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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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Variance of inner product of random vector?

Suppose that we have a random vector $\mathbf{v} \in \mathbb{R}^m$, where each element is sampled from a same distribution of variance $\sigma^2$. Now, we have a constant vector $\mathbf{c} \in \...
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Find the distribution of this Random Variable

i'm stuck with this problem.Can anyone please give me a help? "Find the distribution of a random variable $\mathcal{X}$ such that suppose has the following properties $ m=\mathbb{P( \mathcal{X} = 1})...
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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 $\frac{s}{\sigma} \frac{x}{s} \overset{d}{\to} N(0,1)$ ...
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How can the var(|X|) be defined?

I know that $var(|X|) = E[|X|^2] - (E[|X|])^2 = E[X^2] - (E[|X|])^2$. However, I don’t know if (E[|X|])^2 can be simplified in terms of E[X] or something similar that has no absolute value.
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28 views

Simple expected value and variance exercise

The question is to find the expected value and variance of $X - Y$ where $X, Y$ are independent random variables distributed in $[0,1]$ My Attempt: The expected value is simple enough, where $E(X-Y) ...
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The Cramer-Rao lower bound of $e^{-(x-\theta)}\exp(-e^{-(x-\theta)})$

Let $f(x;\theta ) = e^{-(x_i-\theta)}exp(-e^{-x_i-\theta)})$ How do I find the Cramer-Rao lower bound? the log likelihood is $l(\theta;x)=\Sigma_{i=1}^n{[-(x_i-\theta )-e^{-(x_i-\theta )}]}$ and ...
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33 views

The expectation and variance in case of dependency

I have the following question: We have that $$Z = IB + (1-I)0$$ and $$P(I=1)=q , P(I=0)=1-q$$ Now calculate the expectation and variance of Z and don't assume that B and I are independent. The E[Z]...
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Covariance Matrix of a bivariate normal vector times a constant matrix?

What I have is a bivariate vector W and a constant 6x2 Matrix B, what would the resulting distribution then be of BW? Can it be posed as a multivariate normal vector with a certain Covariance Matrix? ...
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51 views

How to use the Weak Law of Large Numbers to show this? [closed]

Let $X_1,...,X_n$ be an iid (independent and identically distributed) sample with mean $ \mu $ and variance $\sigma^2$. We can use this conclusion :$ (n-1)S^2 = \sum_{i=1}^n (Xi-\overline X)^2 = \...
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How to get this equation from the variance formula?

Let $X_1,...,X_n$ be an iid (independent and identically distributed) sample with mean $ \mu $ and variance $\sigma^2$. How to show $$ (n-1)S^2 = \sum_{i=1}^n (Xi-\overline X)^2 = \sum_{i=1}^n (Xi-\...
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31 views

Maximum Variance of a Distribution

Let the random variable $X$ have the distribution $\mathbb{P}(X=0)=\mathbb{P}(X=2)$, $\mathbb{P}(X=1)=1-2p$ for $0\leq p \leq 1/2$. For what $p$ is the $\mathrm{Var}(X)$ maximum ?
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Variance of a sum of variables with coefficients too

Can someone help me with the proof here? How do I start the proof? How do I simplify $(\sum_{i=1}^k a_i(Y_i - E(Y_i))^2$? I'd end up getting a large multiplication between each $a_i(Y_i - E(Y_i)$ ...
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ls it possible to construct discrete r.v.s given expectation and variance?

Suppose there is a discrete r.v.s X, all we know is: E(X) = 10 and VAR(X) = 2500 Any general way to find PMF of X? Thanks.
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Questions of Variance, mean and interpretation

Let X be a continuous random variable with the density function: $f_x(x) = \begin{cases} x+1, & \text{if}\ -1\leq x \leq 0 \\ -x +1, & \text{if}\ 0 \leq x < 1\\ 0 &...
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31 views

Urn problems: find mean and variance - stuck

I am stuck in a problem, and I can't think of a next step to find the solution. The question is the following: Suppose an urn has $k$ balls, numbered from $1$ to $k$, $k \in \mathbb{N}$. A sample ...
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40 views

Variance of sum of correlated variables

I want to compute the variance of this estimator $\hat{\sigma}^2 = \frac{n}{N}\sum_{i=1}^{N}\big(R_{i} - \frac{1}{N}\sum_{j=1}^{N}R_{j}\big)^2$, where $R_{1}, \ldots, R_{N}$ are i.i.d such that: $ R_{...
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12 views

Variance analysis of the rank of groups of variables

So I am looking to analyse the variance in two groups of random variables. There are 2 sets of 3 random variables A, B and C and X, Y and Z. If I calculate the numerical rank of these variables ...
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21 views

Monte Carlo scenario analysis on ranked data

I’m currently trying to perform a sensitivity analysis on various possible outcomes of a commercial competition. The competition will involve bidders providing a tender which will have a price in ...
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29 views

Explanation behind 'Variance' in statistics

I know there are already some questions asked regarding this, but mine is a little different. I know that variance is calculated to know how spreaded the data is w.r.t mean value. So calculating ...