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.
2,389
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How to compute MAD of a dataset in arbitrary directions?
This is how variance in the direction $d$ is computed for a dataset $X = \left\{ x_1, x_2, \cdots, x_N \,|\, x_i \in \mathbb{R}^M \right\}$, where $\sum^N_i x_i = 0$:
$$ \sigma^2(d) = \sum^N_i \left( ...
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Normal approximation of the ratio of Normally distributed random variables sums
Given $n$ independent Normally distributed random variables $X_i \sim N(\mu_i, \sigma^2_i)$ and $n$ real constants $a_i \in \mathbb{R}$, I need to find an acceptable Normal approximation of the ...
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Expected variance of $N$ random variables chosen from a uniform distribution
Consider the uniform distribution on $[a, b]$. There are $N$ random variables $X_1, X_2, ... , X_N$ chosen from that distribution. How can we determine the expected variance of the $N$ random ...
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$\mathbb E[X^2]$ vs $\mathbb E[X]^2$ in Statistics
In a statistics problem solution it stated:
$$ \mathbb E[\bar{X}]^2 = \operatorname{Var}(\bar{X})+ (\mathbb E[\bar{X}])^2 $$
I remember that
$$ \operatorname{Var}(X) = \mathbb E[X^2] - (\mathbb E[X])^...
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Deriving variance from the expected deviation from the mean of a normal distribution
I know the expected absolute deviation from the mean of a normal distribution $E[|X-\mu_x|]$. From this I want to derive the variance $\sigma^2$ of said distribution. This is done to tune a filter of $...
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Why should a part of the updated posterior variance using kalman filter have a negative sign?
enter image description here
Hi, I am learning the Kalman filter and am really confused at the estimated posterior variance. The conditional posterior variance has a negative term, but to me, it is ...
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Which is the variance of a variable which is the linear sum of normally distributed random variables?
I have a random variable $x$ which is normally distributed with expceted value $\bar{x}$ and variance $\sigma$:
$$x\sim N(\bar{x},\sigma)$$
As you know, i can consider $\bar{x}$ a random variable that,...
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Different ways to arrive at the Covariance matrix
I'm familiar and am comfortable with derivation of the covariance matrix as the sum of squared dot products between data points $x_i \in \mathbb{R}^N$ and a unit direction vector d ($\text{Var}(d) = \...
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How to derive an analogue of the covariance matrix for standard deviation?
The covariance matrix can be interpreted as a summarization of a whole dataset into a single matrix representing a quadratic form that computes the variance of that dataset in a certain direction. I ...
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How does the Covariance matrix encode rotations?
How come the covariance matrix encodes rotation parameters and spread of data? I observed that a covariance matrix for an $N$-dimensional dataset has the following number of degrees of freedom (i.e. ...
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A random sample of $50$ machines obtained that its average life is $\bar{x}=70$ months with a variance of $s^2=49$. Confidence interval for variance.
I need help with the part b) of this exercise.
A random sample of $50$ machines obtained that its average life is $\bar{x}=70$ months with a variance of $s^2=49$. Assume that they are normally ...
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Standard Deviation of 4 Game Series
A game played by B and K involves indepenent rounds. In each round if B wins they receive 1 dollar from K, if K wins they receive 2 dollars from B, and in the event of a draw no money is given. K ...
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Dot product between 2 random vectors
I have 2 independent 2-dimenstional random vectors $A$ and $B$.
$$ A = [a_1, a_2] $$
and
$$ B= [b_1, b_2] $$
The variance of the elements of A are identical ($Var[a_1] = Var[a_2] = \sigma^2_a$).
The ...
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Why doesn't the sample variance become the population variance when the sample has the whole population
We know that for a sample of size $n$ the sample variance is
$\displaystyle S^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}$
Suppose I used the whole population of size $N$ as my sample ...
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Variance of sum of dependent Random Variables [closed]
Let X be a Gaussian random variable, but all X_1 is not independent of X_2, X_2 of X_3, etc. Let Y = sum of all n X's, what is the variance?
So if n = 2, then Var(Y) = Var(X_1) + Var(X_2) + 2 * Cov(...
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Help with solving for variance of estimator
I am struggling with an exercise in estimation theory. I am given the following estimator for the parameter $\theta$:
$r[k] = \frac{x[k]}{\theta} + n[k]$
Where $n[k]$ is zero-mean Gaussian noise.
And ...
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Show that $\mathbb{V}(\bar{X})=\frac{\sigma^2}{n}$
Consider $X_1,\dots,X_n$ to be independent random variables identically distributed (i.i.d) with mean $\mu$ and variance $\sigma^2$.
We have to show that the variance of the arithmetic mean, ...
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Find var$(\overline{X_n}^2)$ given that $EX_1=\mu,E(X_1-\mu)^k=\alpha_k$ [closed]
$X_i\sim F$ are $n$ iid observations. $\overline{X_n}$ is their mean. I want to find var$(\overline{X_n}^2)$ given that $EX_1=\mu,E(X_1-\mu)^k=\alpha_k$.
What I found out are:
var$(\overline{X_n})=\...
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${\rm Cov}(X,Y)=1/4 \times ({\rm Var}(XY)-{\rm Var}(X/Y))$
I saw that formula in this paper, equation (1.3).
The formula should hold for all the random variables such that those functionals are properly defined. I have been trying to prove it, but without ...
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Will the variance of some imaginary variable be negative?
From the formula $$\mathrm{Var}[aX]=a^2\mathrm{Var}[X],$$ if we take $a=\sqrt{-1}$ (so all measurements are pure imaginary numbers or zero), does this make the variance of $\sqrt{-1}X$ possibly ...
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Solving the variance of a compound pdf
Before I state the question, I want to quickly point out that the question is stated using a constant $\sigma$, while also maintaining that the inner pdf $f$ is a mean 0 and variance 1 density. It's ...
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What is ${\rm Var}(X|Y>c)$ given $X$ and $Y$ are jointly normal distributed?
Suppose $X$ and $Y$ are jointly normal distributed with mean $\mu_x$ and $\mu_y$, standard deviation $\sigma_x$ and $\sigma_y$, and correlation $ρ$. What is the conditional variance of $(X∣Y>c)$, ...
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"UMVUE" for a quantity depending on the statistic and unknown parameter
Let $X \sim N(\mu,s_x)$ and $Y \sim N(\mu, s_y)$ be independent Normal random variables with unknown mean $\mu$ and known variances $s_x$ and $s_y$. It can be shown that the statistic $T = tX + (1 - t)...
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Pearson correlation coefficient - proof
Can someone prove this formula ?
standard_devn_second_time_series = sqrt((1 - correlation_coefficient ^ 2) * variance(first_time_series))
first_time_series is given and I need to calculate the ...
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Variance of Variance -> Confidence Interval?
let's consider some random variables collected in the vector $X$ following the distribution $f_X(X)$. We want to compute the probability that:
$$
p = \textrm{Pr} [G(X) < 0]
$$
where $G(X)$ is some ...
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Sum of variances of 2 correlated random variables
I have 2 random variables (RVs) $X$ and $Y$ which are correlated by $\rho$.
$$X, Y \sim N(0, \sigma_1)$$.
X and Y are both multiplied by $Z$ which is also a normal RV, independent from $X$ and $Y$ ($Z ...
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Variance of random matrix?
Let's say $W$ is a symmetric matrix random variable, i.e., $W=W_{ij}$ with probability $P_{ij}$. We already know the definition of $\mathbf{E}[W]$. Is there a definition for $\mathbf{Var}[W]$? And ...
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Expectation and Variance of Infinite Sum
I am presented with the random variable $S = \sum_{k=1}^{\infty} a^{-k} Z_k$ where $a > 1$ and $Z_k$ is defined as $P(Z_k = \pm 1) = 0.5$. As part of an $L^2$ convergence proof, I am tasked with ...
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How does the variance change when a new obervation is added?
Question:
Let $\bar{x}_n$ and $s_n^2$ denote the sample mean and variance. Let $\bar{x}_{n + 1}$ and $s_{n+1}^2$ denote the mean and variance when a new observation $x_{n+1}$ is added. Show that $ n ...
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How can variance be unknown if you know the standard deviation?
Task is to campare two samples, standard deviation is know, variance is unknown (std. deviation is square root from variance) - how is it possible?
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How can I estimate the variance of a dependent variable from a random variable with nonlinear relationship?
I am working on a Kalman Filter and I've added a new state variable that is observable via one of the measurements via nonlinear relationship. My sensor reads $y \in \mathbb{R}$ and I am assuming that ...
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Finding MLE of standard deviation and asymptotic variance of the estimator
Variables $X_{1}, . . . , X_{n}$ - random sample from normal distribution, $N(0,\theta)$. There I need to find:
the MLE of standard deviation of $X_{1}$;
the asymptotic variance of the estimator we ...
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Updating the variance of a sliding window without using stored data
There is a very nice way to compute the variance of a moving window as detailed by Knuth and Cook and answered locally here, also on a blog here. The method requires you to make use of the data in the ...
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Variance of an event that occurs multiple times [duplicate]
You are playing a game with a friend where you flip a coin and if it comes up heads, you give her \$1, and if it comes up tails, she gives you \$1.
If you play the game 10 times, what would be the ...
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Sample variance of a random sample from a normal distribution with mean and variance
I know that if the sample variance of a random sample from a normal distribution $(\mu,\sigma^2)$ is
$$S_1^2 = \frac{1}{n-1}\sum{(X_i-\bar{X})}^2$$
then,
$$U =\frac{(n-1)S_1^2}{\sigma^2}$$ has a $\chi^...
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Upper bounding $\text{Var}(\max_i \{X_i + c_i\})$ using $\text{Var}(\max_i \{X_i \})$
Let $c_i$ be arbitrary positive constants. Is there a way to upper bound $\text{Var}(\max_i \{X_i + c_i\})$ by some constant multiple of $\text{Var}(\max_i \{X_i \})$ that is independent of the values ...
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Can the variance of the sample variance be negative?
In this answer is shown that the variance of the sample variance is
$$
\text{Var}(S^2) = \frac{1}{n} \left(\mu_4 - \frac{n-3}{n-1}\sigma^4\right)
$$
where $\mu_4$ is the fourth central moment, ie $E[(...
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Statistical test for overall standard deviation given covariance matrix
I have a model that predicts values and also gives a standard deviation for the prediction. The standard deviation given depends on the input data for the model and thus is different for every ...
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Variance of the estimators of the first central moments
I want to know what is the variance of the unbiased estimators of the first 8 central moments. The variables are i.i.d. and the distribution is unknown.
Although answering this question seems to ...
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Chi Distribution Simulation Not Giving Expected Variance
I am trying to determine the variance of the l2-norm $(r)$ of $k=9$ normally distributed random variables $z_i \sim N(0,\sigma=0.01)$ by using a Chi distribution with 9 degrees-of-freedom.
However, I ...
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How is realized variance derived for geometric BM?
The realized variance under classical Black Scholes where the stock price process follows a GBM is given as
$$V_T = \frac1T\int_0^T\sigma_s^2ds\qquad (1)$$
however, the texts I have been reading do ...
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Variance of the inner product of independent vectors
Note: I am not asking about the interpretation of covariance as an inner product on the space of random variables.
I have two $n$-dimensional random variables $\vec X, \vec Y\in\mathbb{R}^n$.
Each ...
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what did standard deviation tell us?
in my first course in Statistics when I took the measure of variation the first thing intoduced to me is :(The variance) which has this formula :
\begin{gather*} \sigma^2=\frac{1}{N}\sum_{i=1}^{n}(x_{...
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Finding variance of an estimator
I'm not sure how to express the variance of this estimator. Here's the setup.
We have $X\sim N(0,\sigma^2)$ and want to estimate $\mathbb{E}[\phi(X)]$ where $\phi : \mathbb{R}\to\mathbb{R}$ is some ...
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The variance of the number of runs in the runs up and down test
Consider random permutations of $1,2,\ldots,n$ and let $R=1,\ldots,n-1$ be the number of runs in the permuted sequence. For example, if $n=6$, the sequence
$$
6\quad2\quad4\quad5\quad3\quad1
$$
...
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Show $E(MSE) = \sigma^2$ from a one-way ANOVA
Hello given dataset $Y$ with $a$ factors and $n$ observations. How can I show $E(MSE) = \sigma^2$ from the one-way ANOVA of Y?
Where I'm at,
$$E(MSE) = \sigma^2$$
$$E\left(\frac{SSE}{a(n-1)}\right) = \...
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Expected value and variance of $e^{\frac{2n}{\sum_{i=1}^n X_i^2}}$ (maximum likelihood estimator)
Let $\theta>1$ be an unknown parameter and let $X_1, X_2, ..., X_n$ be a random sample (which means i.i.d. in this case) from the density $f_\theta$ where $$f_{\theta}(x)=x\theta^{-\frac{x^2}{2}}\...
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Variance of ratio of two independent binomial random variables
How can I compute
$$\text{Var}\left[\frac{a+X}{b+Y}\right]$$
for $a, b > 0$, $X\sim Bin(n,p)$, and $Y \sim Bin(m,p)$ and $X,Y$ independent? I know that adding some constant to a binomially ...
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How to prove the following inequality about $Var(x)$?
Let,
$$ X = \frac{1}{n} \sum_{i=1}^n (h\left(s_i,p_i\right) - \mathbb{E}[h])z_i $$ where $(s_i,p_i)$ are input vectors and $z_i \leq B$. I am trying to proof
$Var(X) \leq |h\left(s_i,p_i\right)| \cdot ...
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What is the difference between variance of estimate and estimated variance?
I am in a grad-level statistical inference class and literally getting all my concepts confused. Here's a cutout of the concepts that need most amount of clarifications: Rice Mathematical Statistics ...
