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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35
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3answers
114k views

Determining variance from sum of two random correlated variables

I understand that the variance of the sum of two independent normally distributed random variables is the sum of the variances, but how does this change when the two random variables are correlated?
4
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1answer
1k views

Minimum mean squared error of an estimator of the variance of the normal distribution [duplicate]

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 ...
1
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1answer
476 views

If $m$ tickets are drawn out of $n$ tickets numbered $1$ to $n$, find $V(X)$ where $X$ is the sum of the numbers on tickets.

$m$ tickets are drawn out of $n$ tickets which are numbered from $1$ to $n$. If $X$ denote the sum of the numbers on the tickets drawn. Find $V(X)$. $X = X_1+X_2+...+X_m$ , if $X_i$ can be treated as ...
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3answers
97 views

What is the largest possible variance of a random variable on $[0; 1]$?

What is the largest possible variance of a random variable on $[0; 1]$? It is evident that it does not exceed $1$, but I doubt, that $1$ is actually possible. The largest variance, for which I found ...
3
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3answers
1k views

Unbiased estimator of the variance with known population size

The variance is defined as $$\sigma^2 = \frac{\sum_{i=1}^n (x_i - \bar x)^2}{n}$$ where, $\bar x = \frac{\sum_{i=1}^n x_i}{n}$ If someone wants to estimate this parameter from a sample (s)he must ...
1
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2answers
145 views

Expected value Matrix properties

Let $x\in \mathbb R^3$ be a Gaussian vector with $(\mu_x,\Sigma_y)$, and let $y=Ux$ where $U\in \mathbb R^{3\times 3}$. Find $y$'s expected value and variance as a function of $U$ using $(\mu_x,\...
0
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2answers
41 views

$X\sim\mathrm{Poisson}(\lambda_1 = 5)$ and $Y\sim\mathrm{Poisson}(\lambda_2 = 15)$. Let $Z = X + Y$ . Compute $\mathrm{Corr}(X, Z)$.

Let $X$ and $Y$ be independent random variables such that $X\sim\mathrm{Poisson}(\lambda_1 = 5)$ and $Y\sim\mathrm{Poisson}(\lambda_2 = 15)$. Let $Z = X + Y$ . Compute $\mathrm{Corr}(X, Z)$. Answer: ...
4
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2answers
319 views

Variance of X vs Variance of a binary function of X

Let $X$ be a random variable in $[0, 1]$ and $m$ its median such that $P(X \le m) = P(X \ge m)$. Define $\beta(X)$ as $$\beta (x) = \left\{ \begin{array}{c} \begin{align*} 1&,\space X \ge m; \\...
3
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3answers
943 views

Why is the denominator $N-p-1$ in estimation of variance?

I was recently going through the book Elements of Statistical Learning by Tibshirani et.al. In this book, while explaining the ordinary least squares model, the authors state that assume that $y_i \...
4
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3answers
572 views

The relationship between sample variance and proportion variance?

I'm trying to see the relationship between the sample variance equation $\sum(X_i- \bar X)^2/(n-1)$ and the variance estimate, $\bar X(1-\bar X),$ in case of binary samples. I wonder if the ...
3
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1answer
475 views

Mean and variance of the order statistics of a discrete uniform sample without replacement

In answering calculate the mean and variance of the highest number drawn on lottery based on the lowest number drawn, I couldn't find the mean and variance of the order statistics of a discrete ...
3
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2answers
107 views

Taking the average after randomly knocking out members

I'm looking at a random variable that takes vectors $\newcommand{\vv}{\mathbf{v}} \vv_1, \dotsc, \vv_n \in \mathbb{R}^d$ and calculates their average, after applying "blankout" noise to them. So we ...
2
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1answer
235 views

what is the intuition for the variance of poisson distribution being lambda

poisson distribution: $P(X=x)=\frac{\lambda^x e^{−x}}{x!}$, It is easy to understand the mean the distribution equal to $\lambda$, but what is the intuition for the variance being equal to $\lambda$?
2
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1answer
118 views

Transformation increasing variance of bounded random variable

Consider a random variable $X:\Omega\to[0,1]$ and the transformation to a Bernoulli variable $$ T(X) = \begin{cases}1 &\text{if }X\geq m\\ 0 &\text{otherwise}\end{cases} $$ where $m$ is ...
2
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3answers
46 views

Covariance for stochastic variables

if $X$ and $Y$ are stochastic variables with $\operatorname{Var}(X)=1.34$ and $\operatorname{Cov}(X,Y) = 0.64$, find $\operatorname{Cov}(2X, 3X+2Y)$. No ideas on this one, as I don't see any way of ...
1
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0answers
108 views

Variance of Random Exponential Variables

Just a note: this is a homework question, so feel free to prod me towards the answer if you want :) Also, I'm pretty bad at statistics so sorry in advance if I'm stupid :/ For some $ X_1 ... X_n \sim ...
1
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2answers
290 views

How to find the variance of $U= X-2Y+4Z$? & The Co-variance of $U=X-2Y+4Z$ and $V = 3X-Y-Z$

EDIT If the random variables $X,Y, Z$ have the expected, $$\text{ means: }\mu_{x}=2 \qquad \qquad \mu_{y}=-3 \qquad \qquad \mu_{z} = 4$$ $$ \text{variances: }\sigma_{x}^{2}=3 \qquad \...
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1answer
70 views

Transformation that increases variance of bounded random variable

Consider a bounded random variable $X:\Omega\to[0,1]$. Does there always exist a transformation $T:[0,1]\to \{0, 1\}$ such that $V(T(X))\geq V(X)$? Remarks: $T(X)$ will be a Bernoulli variable with ...
-1
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1answer
50 views

Finding variance using the matrix method [closed]

If the random variables $X,Y$ and $Z$ have means: $$\bar{x}=2 \;\;\; \bar{y}=-3 \;\;\; \bar{z}=4$$ and variances $$\operatorname{Var}(x)=1 \;\;\; \operatorname{Var}(Y)=5 \;\;\; \operatorname{Var}(Z)=2$...
3
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1answer
1k views

Prove that $\min_{a \in \Bbb R} E((X - a)^2) = Var(X)$

Let $X: \Omega \rightarrow \Bbb R$ be a random variable. Prove that $$\min_{a \in \Bbb R} E((X - a)^2) = Var(X)$$ First, we note that $E((X - a)^2) = E(X^2 - 2aX + a^2) = E(X^2) - 2aE(X) + a^2$. ...
3
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1answer
257 views

Why minimising the MSE in Variance-Bias tradeoff?

As I understand the Variance-Bias tradeoff, modifying estimators to minimise bias might increase the variance of the estimator and vice-versa. For the simple case of the biased variance estimator, ...
3
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2answers
242 views

Biased estimator of the Variance of a Gaussian Distribution

I'm reading the Deep Learning book and I've encountered a section that is difficult for me to understand. Specifically transformation from equation (5.38) to (5.39) $$ \mathbb{E}\left[\hat{\sigma}^2_m\...
2
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1answer
87 views

Large variance implies small probability of being in interval

Let $X$ be some random variable on a real interval $[0,\infty)$. Let $[a,b]\subset [0,\infty)$. Intuitively, it seems reasonable that if the variance of $X$ denoted by $\sigma_X^2$ is large compared ...
2
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6answers
392 views

probability variance formula intuition [duplicate]

Here is the formula for the variance : $\sigma^2=\dfrac{\sum(X-\mu)^2}{N}$. My question is why do we SQUARE the difference between the mean and the variable, why don't we use absolute value ...The ...
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2answers
387 views

Variance of squared sum of i.i.d random variables

$X_1, X_2, ..., X_n$ are i.i.d random variables with zero mean and unit variance. Let $X = \sum_{i=1}^n X_i$, I want to express $\mathrm{Var}[X^2]$ in terms of $n$, $\mathrm{E}[X_1^4]$, and constants. ...
1
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1answer
88 views

Standard Error of Coefficients in simple Linear Regression

In the book "Introduction to Statistical Learning" page 66, there are formulas of the standard errors of the coefficient estimates $\hat{\beta}_0$ and $\hat{\beta}_1$. I know the proof of $SE(\hat{\...
1
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1answer
988 views

MLE of variance minimising the mean squared error

While reading up on the derivation of a Maximum Likelihood Estimator for population variance, the MLE for Variance ends up being the biased sample variance, $\frac{n-1}{n}S^2$ where $S^2$ is the ...
1
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1answer
135 views

Variance of alternate flipping rounds

I did the following exercise, but I would like to extend the question to the variance of the variate. Bob and Bub each has his own coin. Chance of coming up "heads" is $\rho$ for Bob's coin and $\...
1
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1answer
79 views

Estimating the number of dice from the number of 6s

Somebody rolls $n$ dice (each has $6$ sides, perfectly balanced) and tells me $k$ of them were $6$s. I'm trying to estimate $n$ based on $k$. I think a good estimate for $n$ is $6k$ (or at least it ...
1
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1answer
66 views

Expected value given probability density function

I am struggling to see the relationship between the left and right side of the equation below: $$\mathbb E\left[\left( e^K \right)^2\right]=\large{e^{2\sigma^2_K}}$$ with $K\sim \mathcal N\left(0,\...
1
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3answers
425 views

Expectation of a conditional variance

I have to prove this $$E(Var(Y|X))=(1-\rho^2)Var(Y)$$ but I got stuck and don't know how to continue. This is what I've done so far based on this variance formula $Var(Y)=E(Var(Y|X))+ Var(E(Y|X))$ $$...
0
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1answer
46 views

Minimum variance of $k_1X+k_2Y$ where $X,Y$ are independent Poisson

I have the following question for homework: Suppose that $X$ and $Y$ are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the combined estimator ...
0
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0answers
79 views

Solving the following maximization problem analytically

Given continuous random variables $x$ and $y$ and a constant $\beta$, define a random variable $z$ by $z:=y+\beta x$. Further, define a random variable $t$ as a function of $z$: $$ t:=z-\frac{A}{2}(z-\...
0
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2answers
239 views

how to interpret the variance of a variance?

I understand how to interpret variance, but, what is the physical meaning behind the variance of a variance? Does it relate to error bounds at all (e.g. confidence interval)?
0
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1answer
1k views

compute confidence interval from “variance of the sample variance”?

(This question spawned from how to interpret the variance of a variance?) For a random variable having a normal distribution with zero mean, we know its sample variance (1.2) and also the variance of ...
0
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1answer
438 views

Probability with covariance matrix

Let $X_1$, $X_2$ be normal with covariance matrix: $$\Sigma = \begin{pmatrix}6&-3\\ -3&7\end{pmatrix}$$ Find the probability that: $P(X_1 - X_2 \ge 1)$ MY ATTEMPT: Let: $Y = X_1 - X_2$ $$\...
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0answers
164 views

Autocorrelation test for Bernoulli distribution

Im running a Bernoulli experiment with probability P and discrete time-intervals. After t=100 I have an average success-rate of 0.6 (60 success and 40 fails) and set P=0.6. (?) Now Im assuming Im ...
0
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1answer
39 views

Null hypothesis testing

enter image description here For this problem, can we simply jump to the conclusion that since the mean of X is 3, and $Y_i=0.17+0.1X_i$, the mean of $Y_i$ is just 0.3?
0
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1answer
87 views

Variance formula, $E^2(x)$ part

To find the variance for a variable I know you're supposed to use: $$Var(X)=E(X^2)-E^2(X)$$ When looking at the solutions to one of my class's problems I see $E(X)=p/(1-p)$. They then go on to say ...
0
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2answers
1k views

Finding Variance from a joint moment generating function

The random vars X and Y have, for all real values of $T_1, T_2$, the joint mgf $M(T_1 , T_2) = \frac{1}{2} e^{T_1 +T_2} + \frac{1}{4} e^{2T_1 +T2} + \frac{1}{12}e^{T_2} + \frac{1}{6} e^{4T_1 +3T_2}$ ...
0
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1answer
70 views

Why variance is not squared before scaling for rolling dice problem?

I am trying to calculate mean $E[Y]$ and variance $Var[Y]$ of sums, when rolling a dice of 6 outcomes $(1,2,3,4,5,6)$, 10 times. I get the mean as below. Mean of sums for 10 tosses $ Y = 10X, \\ \...
0
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2answers
384 views

Stratified Sampling for Variance Reduction--Need Intuition as to Why it Works

When working on variance reduction techniques, I was studying stratified sampling. Suppose we wanted to estimate a definite integral, and we decided to do so using classical Monte Carlo. It can be ...
-2
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1answer
2k views

What is the probability that x is less than 5.92? [closed]

Let x follow a Normal distribution with mean 2 and variance 4. This question is not a duplicate. One question is stating between two variables Basic stats question having trouble figuring this out.