# 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...