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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Expected value of square conditional expected value

I need to prove that $\mathbb{V}(Y)=\mathbb{E}(\mathbb{V}(Y\mid X))+\mathbb{V}(\mathbb{E}(Y\mid X))$. Using the fact that $\mathbb{V}(Y\mid X)=\mathbb{E}\left (\left (Y-\mathbb{E}(Y\mid X)\right )^2\...
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Estimate of the variance of the point estimator [closed]

This is probably a very stupid question, but I tried everything and look all over different topics and forums, i just don't get it. A poll of 1500 registered voters is taken. Of these, 600 say they ...
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21 views

Approximate the gradient of the empirical risk function

Let $D=(x_i,y_i)_{i=1}^n\subset[0,1]^2$ be an iid Dataset where $(x_i,y_i) \sim P$ denotes the joint probability distribution. Let $f^*:X\rightarrow \mathbb{R}$ denote the least square loss Risk ...
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  • 61
3 votes
2 answers
78 views

Variance of product of Gaussian random variables

Suppose I have $r = [r_1, r_2, ..., r_n]$, which are iid and follow normal distribution of $N(\mu, \sigma^2)$, then I have weight vector of $h = [h_1, h_2, ...,h_n]$, which iid followed $N(0, \...
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Find the mean and the variance of $X(1)$ for stochastic differential equation: $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7$

Suppose that $X(t)$ satisfies $\hspace{5cm}$ $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7.$ Find the mean and the variance of $X(1).$ I know that $E[X(1)]$ will result in mean and $E[(X(1))^{2}]$ in ...
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1 vote
1 answer
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Bernoulli Model: Help calculating variance of asset price?

I'm currently looking at the Bernoulli Model for an asset, with spot price $S_{0}$, which can rise to $uS_0$ with probability $p$ or drop to $dS_0$ with probability $q = 1-p$, over a time of $\delta t$...
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-1 votes
0 answers
21 views

Calculate the expectation and variance for normal distribution [closed]

I am solving a question which is about distribution but it is confusing as I am new in the course of applied data analysis. how we calculate the expectation and variance when we have this given ...
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3 votes
1 answer
27 views

Calculating the characteristic function of sum of random variables

I came across this problem in our probability course: Let $\{X_{n}\}$ be a sequence of independent random variables. For all n, $$P\{X_{n}=n\} = P\{X_{n}=-n\} = \frac{1}{2n^2}, \quad P\{X_{n}=1\} = P\{...
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Combine two position estimates with different accuracies

Hi smart people of StackExchange! I have a question regarding the combination of two position estimates (X and Y coordinates) where each position estimate has a different certainty/accuracy. Example I ...
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The integral with functions as bounds.

Let $t = F(x)$ and $x_{\alpha} = F^{-1}(\alpha)$, $x_{1-\alpha} = F^{-1}(1-\alpha)$. Then $$\int_{x_\alpha}^{x_{1-\alpha}} x^2f(x)dx = \int_{\alpha}^{1-\alpha} F^{-1}(t)dt.$$ Where does $x^2$ go? And ...
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What is the variance of a random field multiplied by some fixed function?

I have an ensemble of Gaussian random fields with $X\sim N(\mu, \sigma)$ defined over a set of coordinates $s$. I multiply realisations of this field $x(s)$ by a coordinate dependent function $\psi(s)$...
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Alternating Renewal Process: How to calculate variance without knowing how the two distributions depend on each other

I am trying to solve a Alternating Renewal Process exercise. The "on" state follows a exponential distribution with mean 2. The time in the "off" state follows a gamma distribution ...
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Robust Estimation of Location Parameter. P. J. Huber. (1964).

In the aforementioned book there is a statement: (i) Supremum of the actual variance is infinite for any estimator whose value is always contained in convex hull of the observations. If we take a ...
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Why are these equations describing the variance of residuals equivalent (single-factor model in finance)? [closed]

Residual Equations The above equations describe the residual variance in the single-factor model in finance. I'm struggling to understand why they are equivalent, specifically why the fourth and fifth ...
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Bagging Linear Model?

I have a question regarding bagging linear models. Suppose you wanna do linear regression on data X and y. Alice directly implements (OLS) Linear Regression on it. The model is A1. Bob applies bagging....
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1 answer
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What is Cov(X-Y,Z-W)

I know the equality of the covariance $$ \operatorname{Cov}(X+Y, Z+W) = \operatorname{Cov}(X,Z) + \operatorname{Cov}(X,W) + \operatorname{Cov}(Y,Z) + \operatorname{Cov}(Y,W), $$ But I have the doubt ...
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Variance of predicted value in linear regression model

I am working on a question whereby I have calculated the linear regression model $$Y_i = -20.57 + 1.7446x_i + \epsilon_i ,$$ and have been given the data that $x_i$ = 85. From this, I have calculated ...
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0 votes
1 answer
29 views

Can a sequence of non-eventually-$G$-measurable random variables var-converge to a $G$-measurable random variable?

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $G$ be a sub-$\sigma$-field of $\mathcal{F}$. Suppose that $(X_k)_{k \in \mathbb{N}}$ is a sequence of square-integrable real-...
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Given that $\sqrt{n}(\hat\alpha+\hat\beta-\alpha-\beta)\to_d N(0,\sigma^2)$, what is the smallest possible $\sigma^2$

Let $(X_i)$ $i=1,...,n$ be a sequence of i.i.d random variables from $p_{\alpha,\beta}:\alpha,\beta\in(0,\infty)$ Assume the Fisher information matrix is $I(\alpha,\beta)= \begin{pmatrix} \beta/\alpha&...
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0 votes
1 answer
17 views

Path-wise difference given a perturbation of densities

Let $p$ and $q$ be probability functions, say on $\mathbb{R}$. Suppose that we know $ |p-q|\leq \epsilon $ for some $\epsilon>0$ on $\mathbb{R}$. Let $X\sim p$ and $\tilde{X}\sim q$ be random ...
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  • 173
0 votes
2 answers
23 views

Variance calculation of mean of i.i.d. random Bernoulli variables

Define each i.i.d. indicator variable $X_i$ as Bernoulli with $p = \frac{\pi}{4}$. If we want to find the variance of $X$, which is defined as the mean of $n$ of these indicator variables, then we ...
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26 views

hypothesis testing and properties(biasedness/unbiasedness, consistency) of the OLS estimator

Consider the following equation * \begin{equation} \label{eq:1} C_{t} = \beta_{1} + \lambda Y_{t} + \epsilon_{t} \end{equation} where, \begin{equation} \label{eq:2} E(\epsilon_{t}\mid Y_{t}) = 0 \...
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1 vote
3 answers
39 views

Variance of a random variable in terms of expected value?

When I first encountered the variance of a random variable, I found it in the form: $$\text{Var}(X) = \sum_{i=1}^n (\mu - x_i)^2p_i$$ which is pretty intuitive: it's the sum of each squared distance ...
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1 vote
0 answers
21 views

Need to analytically proof a fomula for variance induction [duplicate]

I am looking into a question about variance induction on an incremental dataset. To begin with, dataset $D_{n-1}$ contains elements $\{x_1, ..., x_{n-1}\}$, and we have got the values of: mean $\bar{...
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15 views

Random variables with same mean and different variance

This is the follow up from question. Suppose $X$ and $๐‘Œ$ with the same distribution, $E[X]=E[Y]$ and $๐‘‰๐‘Ž๐‘Ÿ(๐‘‹)<๐‘‰๐‘Ž๐‘Ÿ(๐‘Œ)$. I know that the argument such that $X$ first-order stochastic dominate ...
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1 answer
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Minimum of the variance of a data set given the variances of subsets

Suppose we have a population data set $X$ which is partitioned into two subsets $A$ and $B$, with population variance $3$ and $4$, respectively. Is it true that the population variance of $A$ is at ...
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Does convergence of conditional variance to 0 imply convergence of unconditional variance to 0?

Let $P := (\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $G$ be a sub-$\sigma$-field of $\mathcal{F}$. Suppose $\{X_k\}_{k \in \mathbb{N}}$ is a sequence of square-integrable real-...
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  • 57
-2 votes
1 answer
38 views

Unbiased estimator from a random sample [closed]

Problem I really don't understand the answer to the below question. It's only 2 marks, so it should be reasonably simple. Attempt I have expanded out the brackets and got as far as $E(Y) = E(X^2) - E(...
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-2 votes
1 answer
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I thought var(a) * var(b) = var(ab) but, it is not? (independent each other) [duplicate]

Let's say that we have two independent variables, r, and g. If two variable is independent, Var(r) * Var (g) should be Var(r*g). However, as shown below, It is not. I do not understand why Var(r*g) is ...
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0 answers
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Determining the sample size to satisfy two tests

A survey of a university's students is to be carried out to estimate both the proportion, $P$, who own a bicycle and the average weekly spend on junk food, $\bar X$. It is desired to estimate $P$ with ...
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1 vote
1 answer
34 views

Variance of piecewise defined random variable

I have the random variable \begin{equation} X = \begin{cases} 0 \;\; &\text{with probability }\frac{1}{2}\\ \exp(\frac{1}{\lambda}) \;\; &\text{with probability }\frac{1}{2} \end{cases} \end{...
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  • 363
0 votes
1 answer
31 views

How come does the variance sum law work for more than 2 independent random variables?

I have learnt through proof that the variance sum law indeed works for two independent random variables, such as var(x + y) = var(x) + var(y). But what I canโ€™t wrap my head around is how this is ...
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-1 votes
1 answer
44 views

Find Var(Y|X=x).

The joint probability density function of X and Y is f(x,y) = 2/3 for 0 < x < 1, 0 < y < 2, x < y, and 0 otherwise. I am trying to use the formula Var(Y|X) = E(Y^2|X)-(E(Y|X))^2. I have ...
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  • 1
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1 answer
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Expectation Value and Variance

I know the formula for Expectation value is $$E(X)=\sum f_ix_i$$ where $f_i$ denotes the PMF(Probability Mass Function) and Variance is $Var(X)=E((X-m)^2)$ where m is E(X). But what is really the ...
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0 answers
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Continous case of Var(E[Y|$x_1$])

I am trying to estimate the variance of the conditional expectation for a continuous case. As an example I use the Ishigami function which is: \begin{align} f(x_1,x_2,x_3) = sin(x_1) + a sin^2(x_2) + ...
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1 vote
1 answer
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Durrett Example 3.4.13 (Infinite Variance)

I have two questions as I read over Example 3.4.13 (Infinite Variance) of Rick Durrett's Probability: Theory and Examples. The example assumes $X_1, X_2, \cdots$ to be i.i.d. R.V.'s and have ...
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  • 459
4 votes
2 answers
74 views

Expected value and variance of number of coin flips until two consecutive tails are flipped

I'm working with a problem from an old exam where one had to calculate the expected value and variance of the number of throws, let's call it $N$, before we get two tails in a row. We also assume the ...
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1 vote
0 answers
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Forecast variance of Time Series model?

I'm reading the book "Forecasting: Principles and Practice" and on the exercise page there are two question, 16 and 17 that I don't know how to answer. Can anyone help how I should start or ...
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0 answers
20 views

Expected value of a transformed random variable Z

Let $Y$ be a random variable. Let $p(x)$ be its probability density function with values given by $$p(1)= p(2) = p(-2) = \frac{1}{5}$$ and $$p(0) = \frac{2}{5} $$ Now define a random variable $Z$ by $$...
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Do the sample coefficients of variation follow a specific distribution when a lot of samples are taken?

I am measuring the coefficient of variation after a process takes place, and I do not know the population distribution. I would like to find the probability the coefficient of variation is smaller ...
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2 votes
1 answer
46 views

When do we add variance and when do we use $Var(cx) = c^2Var(x)$?

This is a problem that I'm working on: A yoga studio is trying to estimate total class sales for next year. They assume that: Between 10 and 14 people attend each class (uniformly distributed) Each ...
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0 votes
0 answers
13 views

Standard error of slope when data is unknown

Letโ€™s say I did a linear regression on a dataset with 10 observations, and I got the estimated slope as $b=0.5$ and residual variance as $\sigma^2=5.2$ and further, $$(A^TA)^{-1}=\begin{bmatrix} 2.54 &...
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1 vote
1 answer
39 views

Variance bounded when values of random variable bounded?

At the risk of asking something dumb, but I'm reading a paper dealing with random variables, say $X$, assumed only to take values in $[0,1]$, have mean $\mu\in[0,1]$ and variance at least $\sigma^2>...
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  • 397
1 vote
2 answers
62 views

what is the variance of difference between max and min of n i.i.d uniform variables : U(0,1)

It is an interview question: calculate the variance of difference between max and min $$variance[\max(\{X_i\}) - \min(\{X_i\})].$$ Here $\{X_i\}$ is n i.i.d uniform variables : U(0,1). I know it is ...
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0 votes
1 answer
33 views

Variance of nested random variables (e.g. dices)

I have a set of independent but not identically distributed dice $D_0, \dots, D_n$, with mean values $\mu_i$ and variances $\sigma_i$. Now i decide to roll one dice at random with probability $p(D_i)$,...
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4 votes
2 answers
110 views

Flajolet & Sedgewick: How to compute the variance of the number of cycles in a random permutation?

I am reading the book Analytic Combinatorics 4ed by Sedgewick and Flajolet. On page 160 at Example III.4 the authors derive the variance of the number of cycles in a random permutation. I can follow ...
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0 votes
0 answers
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Does the population variance equal the variance of a single observation?

According to Wikipedia, the standard error $\sigma^-_x$ of a sample mean can be computed by $\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the standard deviation of a statistical population and $n$ is ...
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29 views

Calculate $E(X^4 )$ and let $s โˆˆ R$ be a constant. Calculate $E(e^{sX}).$

X is a normal random variable such that X~N{ยต, ฯƒ$^2$}. To find $E(X^4 )$ I took $ Y=X^2$ hence $E(X^4)=E(Y^2)=Var(Y)+E[Y]^2=Var(Y)+ (ฯƒ^2+ยต^2)^2$ however I'm unable to find an adequate substitution for ...
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2 votes
1 answer
36 views

Conditional variance of Y given X when y is a continuous function of x

Technically the problem term would be $E[Y^2|X]$ For simplicity, let us for a minute assume that $y = a + bx$ Is it mathematically correct to write: $E[Y^2|X] = E[(a+bx)^2]$, and if so why? And how ...
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  • 358
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0 answers
68 views

Maximizing the variance of weights of Bernoulli RV maximize mutual information?

I have a random variable $X=a_1X_1+a_2X_2 + \ldots a_kX_k$ where $X_i \sim Bern(q)$, $X_i \perp X_j, \forall i,j\in \{1,2\ldots,k\}$. Also $\sum_{i=1}^{k} a_i=k$ and $a_i \in \mathbb{N} \bigcup \{0\...
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