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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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We flip a coin until a tail or five heads in a row occur. What is the number of expected flips?

We flip a coin until a taild or five heads in a row occur. What is the number of expected flips? I have tried to solve this by first defining 2 random variables: X ...
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373 views

Proof of Variance of the Irreducible Error

In Introduction to Statistical Learning, given the general form of a quantitative response between a set of predictor variables and a target variable $$Y=f(X)+\epsilon$$ and the general form for a ...
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Calculating Expectation and Variance

I think I have an idea of how to solve these problems, but I keep getting stuck. Any help would be greatly appreciated! Also, sorry I know I messed up the formatting with the exponents in the problems....
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103 views

Delta Method: Estimate the Variance of $T$

Let $X = (X_1,\ldots,X_n)$ be a random sample, where $X_1 \sim \mathrm{Bern}(p)$. Let $\lambda = e^p$. Question: By law of large numbers, $T=e^{(\bar{X})}$ is a consistent estimator for $\lambda$, ...
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64 views

Expected value and variance of total value of 60 die rolls

I want to find the expected value and variance of the total spots from 60 fair die rolls. The way I tried approaching this is we know that (if x = total value of all die rolls) that $P(x = 60) = (1/...
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789 views

Calculating the expected value and variance of $n$ independent observations of $X$

I am attempting to find the expected value and variance of the random variable $X$ analytically (in addition to a decimal answer). $X$ is the random variable ...
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1answer
140 views

Normalization of standard deviation

I have a question regarding standard deviation. Let me start with an example: I have response times (RT) from users, let's say RT1 = 3s RT2 = 5s RT3 = 8s I have ...
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1answer
394 views

What are the expected value and the standard deviation of the net profit made by the pharmacist on this medicine in any given month?

At the beginning of every month, a pharmacist orders an amount of a certain costly medicine that comes in strips of individually packed tablets. The wholesale price per strip is 100, and the retail ...
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1answer
50 views

Confirmation of the Variance and Standard Deviation result

The following example is about Variance and Standard Deviation copied from Table 6.1 in the book "The Statistical Analysis of Experimental Data" by J Mandel. The Variance given is 667e-8 and ...
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1answer
38 views

Mean variance problem

Let $w=X*r$ with $X$ a real number and $w$ a random variable; let $u=mean(w)-var(w)$. What is the $X$ that gives the maximum $u$? What are the features of $r$ that are reasonable to ask in order to ...
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Under which conditions on $\sigma_1, \sigma_2$ and $\rho_{12}$ the minimum variance portfolio involves no short selling?

If $\rho_{12} \lt 1$ or $\sigma_1 \ne \sigma_2$ then $\sigma_{V}^2$ representing the variance of the portfolio with weights $(w_1, w_2)=(s, 1-s)$ as a function of $s$ attains its minimum value at $$...
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Determine the weights in the minimum variance portfolio.

Consider three securities with the following expected returns, standard deviations of returns, and correlations between returns: $$ \begin{matrix} \mu_1=0.20, & \sigma_1=0.31, & \rho_{12}=\...
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1answer
692 views

Variance of integral

I am trying to understand stochastic calculus and got stuck calculating the following. I need the distribution of a zero bond under the black model, so I am deriving the variance using the second ...
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1answer
31 views

Why does $Var(Y) = Var(\sum_i^n W_iXi) = n(Var(W_i)Var(X_i))$, when $X_i,W_i$ are iid

This was a step in an explanation of Xavier initialization for neural networks. See http://andyljones.tumblr.com/post/110998971763/an-explanation-of-xavier-initialization (around 3/4 the way down). ...
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1answer
38 views

What makes a legal variance matrix?

Straight-forward question: in probability theory, what makes a matrix a 'variance' matrix? The one we have worked with all 'happen' to be symmetric and positive-definite. Are there any more ...
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1answer
490 views

Finite mean implies finite variance proof

I came up with the following proof about what's stated in the title. I know it is wrong but can't find the error. Let $X$ be a random variable with finite mean ($EX<\infty$). The function $f(t)=-t^...
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1answer
212 views

Minimum variance portfolio problem

So the question asks: There are N (N > 1) stocks with the same variance $σ^2$ and the same pairwise correlation coefficient γ (i.e. $c_ij$ = γ for all i = j. γ is a given constant such that 0 ≤ γ < ...
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confused about meaning of a expectation of a function

https://en.wikipedia.org/wiki/Bias%E2%80%93variance_tradeoff#Derivation well,in the "Derivation" part of the wiki link. i don't figure out why $E(f)=f$, does it imply that the function $f$ is ...
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1answer
33 views

Is this a special probability distribution?

Does the distribution function: $\frac{1}{\theta}e^\frac{-y}{\theta} $ Have a special name? If not, how can I find the variance? I keep running into a dead end when I try.
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135 views

Recalculate Standard Deviation

I know how to calculate a standard deviation when given a set of values. You must calculate the mean, then square the difference of each value between the mean to get the variance. Lastly, the ...
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Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal N(\...
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1answer
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Difficulty in finding marginal distribution

Let $X=(X_{1},X_{2})$ have joint pdf.$$f(x_{1},x_{2})=\begin{cases}\frac{e^{-\frac{x_{2}^2}{2}}}{x_{2}\sqrt{2\pi}},\ &\text{if}\ 0<|x_{1}|\le x_{2}<\infty.\\0,\ &\text{otherwise} \end{...
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1answer
499 views

Sample variance: degree of freedom argument

In sample variance we divide by n-1 and not n. I know a couple of arguments for this - one is that this is sort of a normalization to ensure that the expected value of sample variance is equal to ...
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242 views

Find $\text{Var}(N)$ where $P(N = n|Y = y)$ is $\text{Possion}(y)$; $Y$ is a gamma with parameters $(r,\lambda)$

The question is as follows: Suppose that the conditional distribution of $N$, given that $Y = y$, is Poisson with mean $y$. Further suppose that $Y$ is a gamma random variable with parameters $(...
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1answer
713 views

The symbols for variance and covariance

I learned to denote the variance of $x$ as $\sigma_x^2$, and the covariance of $x$ and $y$ as $\sigma_{x, y}$. The covariance of $x$ and $x$ is then $\sigma_{x, x}$, but because that it just the ...
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1answer
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What does “variance unity” mean?: “A normal distribution with mean $\mu$ and variance unity”.

I've just come across the sentence: "Suppose you have the sample $\ldots$ from a normal distribution with mean $\mu$ and variance unity." What does variance unity mean? Variance equals 1? I can't ...
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289 views

Calculating the variance of an average of $N$ iid random variables

I'm having some problems proving that the variance of an average of $N$ iid random variables is equal to $\frac{1}{N}\text{Var}[X_1]$, where $X_1$ is one of the considered random variables. Formally, ...
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25 views

Deriving variance

In kinetic methods of analysis, the rate of appearance of products is often used to infer the initial concentration of a reactant or substrate. The problem with this method is that often the rate ...
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0answers
92 views

Variance of the sum of a random subset [closed]

Let $A$ be the group of whole numbers in the range $[0, n)$. We choose uniformly at random from all subsets of size $k$ ($0 < k < n$). The mean of the sum of the subset is $\frac{n-1}{2} k$. ...
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Variance of Negative Binomial Distribution (without Moment Generating Series)

Given the discrete probability distribution for the negative binomial distribution in the form $$P(X = r) = \sum_{n\geq r} {n-1\choose r-1} (1-p)^{n-r}p^r$$ It appears there are no derivations on ...
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1answer
1k views

Random Variables, Minimize Variance

The variance of $X_1$, $X_2$ are 1 and 4, and the correlation coefficient p=-0.3 1)Calculate the variance of $Z_1 = 2X_1+X_2$ 2)Calculate the value of a that minimizes the variance of $Z_2 = aX_1+(1-...
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425 views

Expected value of mean squares. ANOVA with one factor

currently i'm trying to find the expected value of the mean squares in an ANOVA. However, I think I have a mistake because the latter term is too long and confused me a little. Any help is welcome. ...
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356 views

The variance of a sum of random vectors

There are $n$ vectors each containing exactly $q$ random variables as elements. Each vector is denoted I$_k$. Each variable within the vector has its own (normal) probability distribution, and the ...
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1answer
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Calculating the standard deviation involving a moment generating function

An actuary determines that the claim size for a certain class of accidents is a random variable, X, with moment generation function: $$M_X(t)=\frac{1}{(1-2500t)^4}.$$ Calculate the standard deviation ...
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1answer
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Variance - particular outcomes of $X$

If $x_i$ is a particular outcome of the random variable $X$, and $\mu$ is the mean of the distribution of $X$, then is it true that $E[(x_i-\mu)^2] = Var(X)$? Maybe I should ask a different question: ...
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1answer
4k views

Finding the expectation and variance from a probability generating function

I need some help with the following question. I managed to get the p.g.f., and can get the expectation and variance in the normal ways, but need a helping hand in deducing them through the use of the ...
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How does Variance become an Autocorrelation Function?

"For a Gaussian stochastic process $X=\{X(t)|-\infty<t<\infty\}$ with mean function $\mu(t)=0$ for all $t$, its autocorrelation function is $$ E(X(t)\cdot X(s))=R(h)=\max(0,1-|h|), h=t-s. $$ ...
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291 views

$N$ is a Poisson distribution with mean $4$. Find $\operatorname{Var}(N\mid N \geq 4)$

You are given that $N$ has Poisson distribution with mean $4$. Find $\operatorname{Var}(N\mid N \geq 4)$ I tried to use the definition of variance, where $\operatorname{Var}(X) = E(X^2) - E(X)^2$ ...
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162 views

Monte Carlo integration and variance

With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? $I = ∫^{\inf}_{inf} f(x)p(x) dx$ (with p(x) some ...
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1answer
21 views

Simple variance question: variance of the whole expression when one variable is random

A very simple question about the variance! I'm interested in the variance of an expression with a random variable inside it. For an expression of this form: $a = \sum_{i=1}^{m}(x_i \cdot y_i - z_i)$...
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2answers
279 views

Variance of random walk model?

I'm taking my second term of statistics, and I find myself obsessed with an unnecessary detail...again. As follows: $$Y_t=\rho Y_{t-1}+u_t$$ That is to say, we are working with a time series. We ...
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1answer
164 views

How do you find variance?

Let $\{X_n\}$ be a sequence of independent identically distributed random variables with mean $\mu$ and finite variance. $$T_n= \binom{n}{2}^{-1} \sum_{1\leq i< j\leq n} {X_{i}X_{j}} $$ I have ...
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Variance of a normal distribution for coin toss.

I have difficulties constructing the normal distribution for (20) coin tosses. (Don't ask why, but I never had probability in school.) What is the probability of getting at most 12 heads out of 20 ...
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1answer
213 views

Mean of the deviations from the mean

I ve been struggling to understand the below problem.If I could get help with this problem, it would be greatly appreciated. Consider the data y, y+a, y+2a,….,y+na and the deviations of these ...
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29 views

Three random variables equation

Given the random variables X,Y,Z have the same distribution and fulfil the following equalities: $$Var(X+Y+Z)=21,Cov(X,Y)=Cov(Y,Z)=Cov(Z,X)=1$$ Find $VarX$ and $Var(X+Y)$. I am lost with this ...
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538 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 ...
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1answer
79 views

Calculate the variance and expectection of $\hat{y}$ in a linear regression model

I have the following linear regression model $$y = \beta_0 + \beta_1 \cdot 40$$ where $\beta_0 = 11.1317$, $\beta_1 = 1.01$, and $40$ is simply the value of the predictor variable (I guess). As you ...
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2answers
77 views

Finding Expectation and Variance of $X_1$ and $X_2$

Let $X_1$ and $X_2$ be random variables such that $E(X_i)=μ_i$ and $Var(X_i)=α_i^2$. A. Find $E(X_1+X_2)$ and $E(X_1-X_2)$ in terms of the μ's and α's. B. Suppose that $E(X_1X_2)=α$. Find $Var(X_1+X_2)...
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1answer
1k views

Mean and variance of Binomial Distribution.

I was reading a paper that gives a dynamic programming model of an R&D project. It said that the performance drift (the uncertainty in the performance of the product being developed) follows a ...
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1answer
86 views

Proving the variance of a random variable given R-notation

Here is a picture of my problem Basically, given that $R_X(t) = \log(M_X(t))$, I need to show that $\text{Var}(X) = R′′(0)$. As an attempt, I know that $\text{Var}[X] = E[X^2] - (E[X])^2$ and that $...