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.
-5
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0answers
15 views
Having trouble with given task
The random size is distributed according to the normal law, which is an average of 15. Calculate
this random size dispersion if it is known that the probability of gaining values from the range
[15; ...
0
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1answer
23 views
Prove that $\hat{\sigma^2}=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is not an efficient estimator.
I am asked to show that the unbiased estimator $\hat{\sigma^2}=\frac{1}{n-1}\sum_{i=1}^n (X_i-\bar{X})^2$ is not efficient.
So far I was able to show that the Rao-Cramer Lower Bound is $\frac{2\sigma^...
0
votes
1answer
22 views
Variance of Binomial Distribution - formula
This might be a really stupid question... but still here we go:
For the Variance we have two formulas:
(i)$$
\sigma^2:=\sum_{k=0}^n (k-\mu)^2p = \sum_{k=0}^n (k^2-2npk+n^2p^2)p= (\sum_{k=0}^n k^2- \...
1
vote
0answers
9 views
How to calculate the variance
So i have this question and I wanted to be sure that my calculations were correct.
The standard deviations of the market returns is $0.2(20%)$ and the
covariance between the return on the market ...
1
vote
1answer
32 views
How to calculate the variance of a portfolio?
So I have this question :
Florida Company (FC) and Minnesota Company (MC) are both service companies.
Their stock returns for the past three years were as follows: FC: -5 percent, 15 percent,
20 ...
0
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0answers
7 views
Variance of two random variables that are defined by independent variables
Let $$Λ_1 = \frac{1}{4} Λ^{(2)} + \frac{3}{4} Λ^{(4)},$$
$Λ^{(2)}, Λ^{(4)}$ are independent random variables.
$\mathbb EΛ^{(s)}=\frac{s+5}{s^2}$ and $\mathbb DΛ^{(s)}=\frac{(s+5)^2}{s^2}.$
From the ...
0
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0answers
13 views
transform variance to constant
Suppose I have a sequence of data $\{x_t\}_1^N$ whose variance is a function of time, $\sigma^2(t) = \sigma_0^2 *t$, where $\sigma_0^2$ is a constant. How can I transform the variance of the entire ...
1
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0answers
6 views
Proof of Fermi Estimation Variance
On the wikipedia page https://en.wikipedia.org/wiki/Fermi_problem, it claims that
"In continuous terms, if one makes a Fermi estimate of n steps, with standard deviation σ units on the log scale from ...
-1
votes
2answers
27 views
Barttletts test and F test with 3 variables [on hold]
I linked a picture with the assignment.
I know how to do F-tests with 2 variances and means, but with 3, I am out of luck. I could do it if I had a dataset in R. but not manual in R or by hand, as it ...
1
vote
0answers
16 views
Variance service times M/G/c queue
I am wondering about the influence of the variance of the service times in an M/G/c queue on the probability that a customer has to wait. Intuitively, I would say that smaller variance implies smaller ...
0
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1answer
34 views
Given $E[X] , \operatorname{Var}(X)$ and $Y\mid X \sim U(X,1)$, find $E[Y]$ and $\operatorname{Var}(Y)$
For $X, Y$ random variables, given $E[X] = \mu$ ; $\operatorname{Var}(X) = \sigma^2$; $Y\mid X \sim \text{Unif}(X,1)$: Find $E[Y]$ and $\operatorname{Var}(Y)$.
(1) To find E[Y], I used the law of ...
1
vote
0answers
22 views
Calculate p-value between two lists of floats of unequal size
I would like to calculate the degree of variance between to lists of floats of unequal size expressed in a p-value.
I tried a two-sided t-test as in the example below using python. But answers that ...
0
votes
3answers
39 views
Why are there two formulas for the sample variance?
I'm using an introductory statistics textbook and it mentioned these two formulas for the sample variance:
$s^2 = \frac{\sum(x - \bar{x})^2}{n - 1}$
and
$s^2 = \frac{\sum{}x^2 - \frac{(\sum{}x)^...
1
vote
1answer
25 views
Probability $P(\mu - d\sigma < X < \mu +d\sigma) \ge 1-\frac 1 d^2$
Show, that $P(\mu - d\sigma < X < \mu +d\sigma) \ge 1-\frac 1 d^2$ , when $\mu < \infty$ and $\sigma^2>0$ and $ d>1$
My Idea:
$P(\mu - d\sigma < X < \mu +d\sigma)$ = $1-P(X \ge \...
0
votes
0answers
14 views
Deriving bootstrap variance of absolute central moments
My statistics text introduces the bootstrap as follows:
Draw $X_1^{*},\ldots,X_n^{*} \sim \hat{F}_n$.
Compute $T_n^{*} = g(X_1^{*},\ldots,X_n^{*})$ where $g$ is any function you want to ...
0
votes
0answers
11 views
Signal-to-noise ratio
Consider the linear structural equation model with known $\beta$ as
$$ SEM \quad X_k = \sum_{j=1}^{p}\beta_{jk}X_j + \epsilon_k$$
where $X_k$ is a random variable. I construct a data matrix $D_{m \...
2
votes
3answers
50 views
Probability- 6 digit number that is built from the numbers 2,5,6,9
A random 6 digit number is picked that is built only from the numbers $2, 5, 6, 9.$
What is the probability that the number can be divided by $3$?
What is the probability that the number can be ...
0
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0answers
21 views
Find covariance matrix of $\frac{f(x + y) }{x + y}$ function
The conditions are the same, but my task is to find covariance matrix. I only noticed that density function is symmetric so expected values, variance are also the same. But I don't know how to find ...
1
vote
3answers
34 views
Calculating Variance(X + Y + 1)
Not Duplicate. I know a question with similar data has been used here, but I am looking for something else.
Two tire-quality experts examine stacks of tires and assign a quality rating to each tire ...
0
votes
1answer
21 views
Measuring the uniformity or closeness of a set of given values
Say we have a sample space of size $N$ . What's a good way to measure how close the values are to one another? In other words, to what degree are the values 'equal' to one another? I thought of a ...
0
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0answers
17 views
$Var(\frac{1}{nh^d}\sum_{i=1}^n Z_i)=\frac{1}{n}E\bigg[\big(\sum_{i=1}^n Z_i-E(Z_i)\big)^2\bigg]$
Let $\{Y_i,X_i\} \in \mathbb{R}\times\mathbb{R}^d$ be a strictly stationary sequence of random vectors and consider $\hat{\Psi}(x)=\frac{1}{nh^d}\sum_{i=1}^nY_iK(\frac{x-X_i}{h})$, where $h=o(1)$ is ...
0
votes
1answer
14 views
Sequence of independent random variables with same expected value such that the weak law doesn't hold
I'm looking for a specific counterexample for the weak law of large numbers. That is, I want a sequence of random variables with same finite expected value $\mu$. These random variables must each have ...
1
vote
0answers
50 views
Which is faster: a bank with five lines of ten or one line of fifty?
I'm working on a probability question with mean and variance.
Let's say that I have two banks. They are identical in every way,
except that bank A has five lines with ten people and bank B has ...
3
votes
0answers
39 views
Help me find my mistake for this variance
Suppose X is an observation from a distribution with probability mass function
$$X\sim f(x)=\left(\frac{\theta}{2}\right)^{\left | x \right |}(1-\theta)^{1-\left | x \right |} 1_{A}(x)$$ $$0<\...
0
votes
1answer
18 views
What is the standard deviation of the frequency table below?
Hello, I have some issues with finding the standard deviation of this question.
I found the mean as 337.5 km, and I considered that standard deviation is the root of the variance.
I first squared the ...
0
votes
1answer
26 views
Estimate variance in a nested study design
I have a "mother" bacteria. I cloned it $R$ times and measured a variable $P$ on these $R$ clones and recorded the mean $\bar p$ and the variance $V$ among these clones.
Then, I made $M$ mutants ($M$ ...
3
votes
1answer
43 views
Proving $V(X) = E(V(X|Y)) + V(E(X|Y))$ using the pythagorean theorem
I know the textbook proof of $$V(X) = E(V(X|Y)) + V(E(X|Y))$$ but I'm interested in understanding the weird proof/analogy with the pythagorean theorem my professor gave in class.
With $X, Y$ random ...
0
votes
0answers
21 views
Variance of random variable (Langevin Equation)
I am trying to solve the following problem and I am slightly confused/stuck.
The Langevin equation reads:
$$\gamma \frac{d x(t)}{d t} = -\frac{\partial V(x)}{\partial x} + \zeta(t) \tag{Eq. 1}$$
...
2
votes
1answer
58 views
Why are there two formulas for variance of random variables?
I'm using an introductory statistics textbook and it mentioned this:
Definition: If $X$ is a random variable with mean $E(X) = \mu$, then the variance of $X$ is defined by $Var(X) = E((X−\mu)^2)$....
0
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1answer
14 views
Variance of an estimator
I was wondering how to compute the variance of the following estimator for the mean. Where $\{Y_1,Y_2,Y_3\}$ is a sample from an exponential distribution :
$$\hat{\theta} = \frac{Y_1 + Y_2}{4} + \...
0
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0answers
9 views
How to determine the variance of this Bayesian linear regression model
I am struggling with the following problem. I have a Linear regression model which uses bayesian statistics: $X_7 = X\beta + \epsilon$. Here $X = (X_0,X_1,...,X_6)$ is a data frame containing 7 ...
1
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0answers
41 views
Value At Risk Elliptical Distributions
I thought I understood elliptical distributions, but then I staggered over the following problem:
Let d financial returns be modeled as the components $X_1,...X_d$ of a d-dimensional random vector $X$...
0
votes
3answers
44 views
How to calculate covariance with an minus like V(X-Y)?
my task is this:
Be $ X $ and $ Z $ independent with the same distribution and $ Y :=X-Z . $ Calculate $ \operatorname{cov}(X, Y) $
and $ \operatorname{corr}(X, Y) . $
My Problem is the minus in $...
0
votes
1answer
16 views
Law of total variance and covariance given X and Y are normal
I have a problem which asks me to find $\Bbb E[Y]$ and $Var(Y)$ given that $Y\text{~}Normal(x,1)$ conditional on $X=x$. $X$ is standard normal. So I have worked out that $\Bbb E[Y]=0$ using the law of ...
0
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1answer
38 views
How to calculate E[Xi Xj]?
This question is from an example in the book of Bertsekas. (p240 of 1st edition). I would like to know why $$E[X_{i} X_{j}] = P(X_{i} = 1\text{ and }X_{j}=1)$$ and $$E[X_{i}] = P(X_{i}=1)$$. please ...
-1
votes
1answer
14 views
Variance of inverse gamma distribution
Given a random variable $X$ which is distributed gamma with shape $\alpha$ and rate $\lambda$, for which the variance is known, how does one calculate $\text{Var}(\frac{1}{X})$? I am hoping not to ...
1
vote
1answer
23 views
DEMONSTRATION FINITE-SAMPLE PROPERTIES OF LEAST SQUARES $\frac{(N-k)S^2}{\sigma^2}\sim\chi^2[n-K]$
Im a Student of Economics, and I have a concern. In the solution of
$\frac{(n-K)S^2}{\sigma^2}\sim\chi^2[n-K]$
How can I show that if the matrix is symmetric and idempotent between
$(I-H)=|| (I-...
0
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1answer
19 views
Difference of normal random variables
I have two random variables $$X_{s+t} \sim N(0, s+t)$$ $$X_s \sim N(0, s)$$ where $s \leq t$. How do I show that...
$$X_{s+t} - X_s \sim N\left(0,s + t + s -2\sqrt{s(s+t)} \right)??$$
I understand ...
1
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2answers
43 views
How to calculate the bias of the estimator for variance?
Question: For observations $x_1$, $x_2$, . . . , $x_n$ with sample average $\bar{x}$, we can use an estimator for the population variance:
$\hat\sigma^2$ = $\frac1n\cdot $ $\sum\limits_{i=1}^n (x_i - ...
0
votes
2answers
43 views
How to use the law of total variance
I know that the law of total variance states
$$Var(X)=\Bbb E[Var(X|Y)]+Var(\Bbb E[X|Y])$$
But how does one treat $Var(X|Y)$ and $\Bbb E[X|Y]$ as random variables? For example, say we know that $$\Bbb ...
1
vote
1answer
52 views
$Y = \frac{X_1 X_2}{X_3}$ where $X_i$ is a uniform random variable
$Y = \frac{X_1 X_2}{X_3}$ where $X_i\sim U(0,1)$ and $X_1,X_2,X_3$ are i.i.d
I need to calculate $Var(Y)$ and $Var[Y|X_3=1.7]$
I know that for each $X_i$,
$E[X_i]=\frac{1}{2}$
$Var[X_i]=\frac{1}{...
0
votes
0answers
25 views
Proving expectation and variance of a function of a random variable tends to a fix point
Given $f:\mathcal{X} \rightarrow \mathbb{R}$ is a continuous function and $\mathbb{E}_{Q(X)}[X] \rightarrow x^\star$ ($x^\star$ is a fix number), $\mathbb{V}\text{ar}_{Q(X)}[X] \rightarrow 0$. How can ...
0
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0answers
16 views
Proving convergence of expectation and variance given Rényi's $\alpha$-divergence tends to 0
I denote $p, q$ as density function of $P, Q$.
Given $Y, X$ are random variables and
\begin{align}
\int q(x)\mathbb{D}_{\alpha}[p(Y\mid X=x)\,||\,p(Y\mid x^\star)] \,dx \rightarrow 0
\end{align}
...
0
votes
2answers
24 views
Can Conditional Expected Value be negative in normal distribution?
So, the problem gives me this facts (for a Normal bivariate distribution X,Y)
$$Var(Y|X=x) = 5$$
$$E(Y|X=x) = 2 + x$$
It asks me to find $$E[Y^2|X=7]$$
I tried this: using the conditional variance
...
1
vote
1answer
17 views
For a sequence of experiments where each $X$ is the number of trials until success with varying $p$, is each $X$ independent?
Assume that, every time you buy a box of Wheaties, you receive a picture of
one of the $n$ baseball player. Let $X_k$ be the number of additional boxes you have to buy, after you have obtained $k-1$ ...
0
votes
1answer
16 views
Finding the probability of success that maximizes the variance of independent trials
A professor wishes to make up a true-false exam with n questions. She assumes that she can design the problems in such a way that a student will answer
the jth problem correctly with probability $...
0
votes
1answer
12 views
Find mean and variance from mgf where t is denominator
For continuous random variable X,
pdf: $f_{X}(x)=2(1-x), x\in[0,1]$
mgf: $M_{X}(t)=\frac{2(e^t-t-1)}{t^2}$
Problem is to find mean and variance from mgf, I tried using $\frac{d}{dt}M_{X}(0)$ and $\...
2
votes
1answer
29 views
Rao-Blackwell and Cramer-Rao LB comparison
Let $X_1, X_2, \dots, X_n$ be a random sample following the Geometric distribution.
$$
\prod\limits_{i=1}^{n} f(x_i|p) = (1-p)^{\sum\limits_{i=1}^n x_i-n}p^n
$$
Since the pmf of the Geometric ...
0
votes
2answers
34 views
Difficult demonstration - How to show that $H_n$ is normal distributed $N(\xi,\sigma^2)$ starting from its moments $ξ$ and $σ$?
I was thinking that if the function $H_n$ of cumulative distribution converges to a distribution $H$, then $\epsilon_n$ should converge to $\epsilon$ what could be expressed as follows:
If $H_n$ is ...
1
vote
1answer
41 views
Law of total covariance inequality
The law of total variance: $$ \text{Var}(X) = \mathbb{E}(\text{Var}(X\mid Y)) + \text{Var}(\mathbb{E}(X\mid Y)).$$
There is also something called the law of total covariance: $$ \text{Cov}(X,Y) = \...
