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.
0
votes
1answer
26 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
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0answers
18 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 ...
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0answers
12 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}
...
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2answers
20 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
...
0
votes
0answers
10 views
Weibull distribution -variance, expected value [on hold]
If $X \sim Exp(\lambda)$, so what is a variable distribution of $Y=X^{\frac{1}{\alpha}}$ for $\alpha >0$? Calculate the expected value, variance and intensity of failures.
For which the value of $\...
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
19 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 ...
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votes
2answers
27 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
37 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: $$ \text{Cov}(X,Y) = \...
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votes
0answers
16 views
Variance of linear combination
This is a follow up question to this.
Let $(X_1,\ldots, X_n)$ be non-independent random variables such that
$$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$
where $\mathcal{N}(0,1)...
0
votes
1answer
51 views
Why is the standard deviation described as $\sqrt{pqn}$ sometimes and sometimes as $\sqrt{\frac{pq}{n}}$?
I assume it has something to do with whether we start with a distribution or with samples, but why is the standard deviation increasing with $n$ in one case and decreasing in the other?
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0answers
6 views
Mean and Variance in the Jacobi Stochastic Volatility model
I would like to compute $ E[X_{T}]$ and $Var[X_{T}]$ in the Jacobi model, where the Dynamics are given as
$ dY_{t}=\kappa(\theta-Y_{t})dt+\sigma\sqrt{Q(Y_{t})}dW_{1t}$
$dX_{t}=(r-\delta-0.5Y_{t})dt+\...
0
votes
1answer
38 views
Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$
I am working on this problem.
Find a consistent estimator for $E[X^2]$ when $X \sim \text{Exp}(\beta)$ .
So far I am thinking of using the invariant property of MLEs, so I let
$$\hat{\theta} = \...
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votes
0answers
7 views
Determining if 2 variances are equal with a rule of thumb?
So yes I am familiar with the F-test and know how to use it. Though I remember there was a quick rule of thumb of determine if 2 variances are equal. By either subtracting or dividing the 2 and it ...
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0answers
10 views
Mean and variance of conditional mean and variance
I was given two discrete r.v. $X$ and $Y$.
I know how to compute $E(X|Y=y)$ and $V(X|Y=y)$ and i realize how you could treat $E(X|Y)$ and $V(X|Y)$ as random variables them self.
Yet I'm ...
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votes
0answers
6 views
Reference relating variance and covariance to elementary linear algebra for undergraduates
I'm currently teaching a non-calculus probability and statistics course for business students, and I have a student who is interested in why we use the variance instead of the Mean absolute deviation(...
2
votes
2answers
115 views
Variance and covariance inequality
Given a real-valued random variable $X$, is
$$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$
true?
Any pointers for how to tackle this problem would be immensely helpful.
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0answers
17 views
Does measuring the variance above the mean only give you a better indication of ceiling/potential?
I want to find NBA player's who have a better chance of scoring an extremely high number of points based on their average and variance. However, with total variance you get numbers that also fall ...
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0answers
21 views
Trouble with seemingly extremely simple statistics question involving normal distribution and expectation.
Say we have a random variable X which is distributed like such: X ~ N(1, 4).
A question asks me to calculate $E(X^2)$, which I thought would be straight forward. I use the formula for Variance: $Var(...
1
vote
1answer
30 views
Calculating variance of a sum
The number of students per day has the distribution N ∼ Poisson(10).
The students of CSUEB withdraw money from a cash machine according to the following probability function (X):
X | 50 | 100 ...
2
votes
1answer
15 views
Show that the autocovariance function depends on $s$ and $t$ only through their difference $\left|s-t\right|$
Consider the time series
$$
x_t = \beta_1 + \beta_2t + w_t,
$$
where $\beta_1, \beta_2$ are known constants and $w_t$ is a white noise process with variance $\sigma^2_w$.
I want to show that the ...
0
votes
1answer
40 views
Inequality for Uniform Distribution
Let $X_1,..,X_n$ be a random independent sample $X_1,…,X_n$ from a Uniform$[0,\theta] $ distribution, $\theta \in [0, \infty)$, with probability density function
$f(x;\theta) =
\begin{cases}
1/\theta,...
0
votes
1answer
24 views
mean for cumulative distribution function
I'm trying to find the mean (expected value) and variance for the following distribution function:
$F(x)=\begin{cases}
0 & \text{for } x \lt 1\\
\frac{x^2-2x+2}{2} & \text{for } 1 \...
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0answers
13 views
Variance of Residuals in Multiple Linear Regression
If I have $n$ variables $x_1,\dots,x_d$ in my dataset, and I regress the first one against all the others i.e.
$x_1=a_{12}x_2 + a_{13}x_3 + \dots + a_{1d} x_d + \epsilon_1$,
how can I calculate $\...
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votes
2answers
29 views
What is the expected value and variance of a random variable that is based on another random variable?
Im unsure if this is the correct question, sorry.
Let B be the a random variable with expected value 10 and variance 4 that is defined on B >= 0.
If Y = 4 + B/10, what is the expected value and ...
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votes
0answers
23 views
Calculate expected value and variance of a t-student distribution without using density function
Let $(X_n)_n$ a suite of random variables independent and identically distributed, $X_i \sim \mathcal{N}(0,1)$ and let $Y_n:= \sum_{j=1}^n X_j ^2 \sim \chi^2_n$ a chi-square random variable with $n$ ...
0
votes
1answer
26 views
Conceptual difference: infinite mean and non-existent mean
Let $X$ be a random variable with finite variance.
I know that this implies a finite mean.
Do I have to prove that or does it follow from the definition of the variance?
The variance is defined as
$...
2
votes
1answer
37 views
Finding Variance of Piecewise Function of Two Random Variables
I have a piecewise function of two random variables:
$$h(X,Y) = \left\{
\begin{array}{ll}
kXY \qquad \text{if } X\geq a\\
kaY \qquad \text{ if } X < a\\
...
1
vote
0answers
22 views
Sum of exponential series of equal mean and variance
Assuming $A$ and $B$ are two non-negative real-valued random variables such that
$\mathrm{E}(A)=\mathrm{E}(B)$ (equal means)
$\mathrm{Var}(A)=\mathrm{Var}(B)<\epsilon$ (equal small variances)
is ...
0
votes
1answer
12 views
Variance of inner product of random vector?
Suppose that we have a random vector $\mathbf{v} \in \mathbb{R}^m$, where each element is sampled from a same distribution of variance $\sigma^2$.
Now, we have a constant vector $\mathbf{c} \in \...
1
vote
1answer
36 views
Find the distribution of this Random Variable
i'm stuck with this problem.Can anyone please give me a help?
"Find the distribution of a random variable $\mathcal{X}$ such that suppose has the following properties
$ m=\mathbb{P( \mathcal{X} = 1})...
2
votes
0answers
48 views
Why can we use a consistent variance matrix estimator when finding the asymptotic distribution
So I am pretty sure that in a one-dimensional case, we would just say $x \overset{d}{\to} N(0,\sigma^2)$ and $s^2 \overset{p}{\to} \sigma^2$ so $\frac{s}{\sigma} \frac{x}{s} \overset{d}{\to} N(0,1)$ ...
1
vote
2answers
16 views
How can the var(|X|) be defined?
I know that $var(|X|) = E[|X|^2] - (E[|X|])^2 = E[X^2] - (E[|X|])^2$.
However, I don’t know if (E[|X|])^2 can be simplified in terms of E[X] or something similar that has no absolute value.
2
votes
2answers
28 views
Simple expected value and variance exercise
The question is to find the expected value and variance of $X - Y$ where $X, Y$ are independent random variables distributed in $[0,1]$
My Attempt:
The expected value is simple enough, where $E(X-Y) ...
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0answers
49 views
The Cramer-Rao lower bound of $e^{-(x-\theta)}\exp(-e^{-(x-\theta)})$
Let $f(x;\theta ) = e^{-(x_i-\theta)}exp(-e^{-x_i-\theta)})$
How do I find the Cramer-Rao lower bound?
the log likelihood is
$l(\theta;x)=\Sigma_{i=1}^n{[-(x_i-\theta )-e^{-(x_i-\theta )}]}$
and ...
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votes
0answers
33 views
The expectation and variance in case of dependency
I have the following question:
We have that
$$Z = IB + (1-I)0$$
and $$P(I=1)=q , P(I=0)=1-q$$
Now calculate the expectation and variance of Z and don't assume that B and I are independent.
The E[Z]...
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votes
0answers
11 views
Covariance Matrix of a bivariate normal vector times a constant matrix?
What I have is a bivariate vector W and a constant 6x2 Matrix B, what would the resulting distribution then be of BW? Can it be posed as a multivariate normal vector with a certain Covariance Matrix? ...
0
votes
1answer
51 views
How to use the Weak Law of Large Numbers to show this? [closed]
Let $X_1,...,X_n$ be an iid (independent and identically distributed) sample with mean $ \mu $ and variance $\sigma^2$.
We can use this conclusion ：$ (n-1)S^2 = \sum_{i=1}^n (Xi-\overline X)^2 = \...
1
vote
1answer
39 views
How to get this equation from the variance formula？
Let $X_1,...,X_n$ be an iid (independent and identically distributed) sample with mean $ \mu $ and variance $\sigma^2$.
How to show $$ (n-1)S^2 = \sum_{i=1}^n (Xi-\overline X)^2 = \sum_{i=1}^n (Xi-\...
0
votes
1answer
31 views
Maximum Variance of a Distribution
Let the random variable $X$ have the distribution $\mathbb{P}(X=0)=\mathbb{P}(X=2)$, $\mathbb{P}(X=1)=1-2p$ for $0\leq p \leq 1/2$. For what $p$ is the $\mathrm{Var}(X)$ maximum ?
0
votes
1answer
10 views
Variance of a sum of variables with coefficients too
Can someone help me with the proof here?
How do I start the proof? How do I simplify $(\sum_{i=1}^k a_i(Y_i - E(Y_i))^2$?
I'd end up getting a large multiplication between each $a_i(Y_i - E(Y_i)$ ...
0
votes
1answer
10 views
ls it possible to construct discrete r.v.s given expectation and variance?
Suppose there is a discrete r.v.s X, all we know is:
E(X) = 10 and VAR(X) = 2500
Any general way to find PMF of X?
Thanks.
1
vote
1answer
22 views
Questions of Variance, mean and interpretation
Let X be a continuous random variable with the density function:
$f_x(x) =
\begin{cases}
x+1, & \text{if}\ -1\leq x \leq 0 \\
-x +1, & \text{if}\ 0 \leq x < 1\\
0 &...
0
votes
1answer
31 views
Urn problems: find mean and variance - stuck
I am stuck in a problem, and I can't think of a next step to find the solution. The question is the following:
Suppose an urn has $k$ balls, numbered from $1$ to $k$, $k \in \mathbb{N}$. A sample ...
0
votes
0answers
40 views
Variance of sum of correlated variables
I want to compute the variance of this estimator $\hat{\sigma}^2 = \frac{n}{N}\sum_{i=1}^{N}\big(R_{i} - \frac{1}{N}\sum_{j=1}^{N}R_{j}\big)^2$, where $R_{1}, \ldots, R_{N}$ are i.i.d such that: $ R_{...
0
votes
0answers
12 views
Variance analysis of the rank of groups of variables
So I am looking to analyse the variance in two groups of random variables.
There are 2 sets of 3 random variables A, B and C and X, Y and Z. If I calculate the numerical rank of these variables ...
0
votes
0answers
21 views
Monte Carlo scenario analysis on ranked data
I’m currently trying to perform a sensitivity analysis on various possible outcomes of a commercial competition.
The competition will involve bidders providing a tender which will have a price in ...
0
votes
1answer
29 views
Explanation behind 'Variance' in statistics
I know there are already some questions asked regarding this, but mine is a little different. I know that variance is calculated to know how spreaded the data is w.r.t mean value.
So calculating ...
