All Questions
Tagged with variance probability
232 questions
2
votes
1
answer
120
views
How to calculate $Var(X)$ for the union size?
Let's say there is a set of $4$ bags $\{{a,b,c,d\}}$ containing balls of colors $\{{red,blue,green,orange,black\}}$. Balls are assigned to bags in an arrangement that allows a variable number of ...
- 113
2
votes
1
answer
79
views
Bartholomew estimate of variance of median for exponential distributed RVs
Suppose, $X_1, X_2, \ldots X_n \sim \text{iid Exponential}(\theta)$. The median is given by $\log(2) \theta$. The MLE of the median is given by:
$$ \hat{M} = \log(2) \sum_{i=1}^n X_i / n $$
And the ...
- 64.8k
2
votes
1
answer
499
views
Buffon's Needle problem
So I'm working through some computational stats stuff from a free pdf of a book. Specifically I'm looking at their take on the classic Buffon's needle problem. The question has a theoretical part and ...
- 121
2
votes
1
answer
72
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How to set $\alpha,\beta$ such that $logit^{-1}(\alpha X_1+\beta X_2)$ has a mean of 0.4 with $X_1 \sim Bern(p)$ and $X_2\sim N(\mu,\sigma^2)$?
I am working in R, and am trying to generate values of
$$
logit^{-1}(\alpha X_1+\beta X_2)
$$
with $\alpha,\beta$ such that $logit^{-1}(\alpha X_1+\beta X_2)$ ...
- 4,260
2
votes
1
answer
77
views
Given , $X$ is a standard normal R.V , I know $E[X|X>c]$ = $\frac{\phi(c)}{1 - \Phi(c)}$ , how do i derive a similar formula for $var[X|X>c]$
I can derive $E[X|X>c]$ = $\frac{\phi(c)}{1 - \Phi(c)}$ , using the trick $- \int \frac{d \phi(x)}{dx} = \int x \phi(x) dx$. How do I do a similar thing to derive $var[X|X>c]$.
- 31
2
votes
1
answer
191
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what is the linear minimum mean squared estimator for y given x of the shaded region?
A 2D random point (x,y) is uniformly distributed on the shaded region of the figure.
What is the linear MMSE estimator for y, given x?
This is what I have so far:
Since it's a linear estimator, I ...
- 917
2
votes
1
answer
883
views
Confusion about the meaning of unexplained variance in R2 interpretation
I want to think about $R^2$ as (in the context of forecasting with different models):
$$\frac{\text{explained variance}}{\text{total variance}} = 1 - \frac{\text{unexplained variance}}{\text{total ...
2
votes
1
answer
818
views
Horvitz-Thompson variance estimation when estimating across strata
I have a sample of Business units, which has been stratified according to two stratification variables (Revenue class and field of Business acitivity). Within the strata, Units were sampled according ...
- 97
2
votes
1
answer
127
views
Calculating the error or variance in $p$ when fitting a binomial distribution to data
I have data fitting a binomial distribution with $n$ total observations and m positive observations. My estimate of $p$ is $\frac{m}{n}$, but is there a way I can estimate the error or variance in $p$?...
- 628
2
votes
0
answers
32
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Proportion of explained variance for a probability model(binary logistic regression)
in the article written by Frank Harell ,Statistically Efficient Ways to Quantify Added Predictive Value of New Measurements,(https://www.fharrell.com/post/addvalue/)
Harell is writing:
For a ...
- 1,035
2
votes
1
answer
239
views
Will change in standard deviation impact covariance?
If we increase the degree of standard deviation of one variable, does it affect covariance of two variables?
Example, two variables are there, A & B, the covariance of A & B is 100, and the ...
2
votes
1
answer
225
views
Can the variance of a U-statistic be of the order $O(\frac{1}{n^2})$?
It is not that easy to find estimators $T_n$ such that $\mbox{Var}[T_n] \sim O(n^{-B})$ with $B = 2$. In most cases, $B=1$.Here $n$ is the sample size. It seems, according to this paper on U-...
2
votes
0
answers
56
views
Variance of 2 Protocols: Sampling Coloured Balls with Dots
Suppose, we have an urn where each ball has one of $M$ colours and some balls have a dot. We would like to estimate the proportion $p$ of balls that have a dot. We have two experimental protocols:
We ...
- 21
2
votes
0
answers
234
views
Calculate Variance from Dirichlet-like Distribution Empirically
I'm interested in the proportion of time that a sensor is in a particular state. The sensor tells me the amount of time that it's in each state, which I will denote by $X = \{ X_1, X_2, X_3\}$. I ...
- 737
2
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0
answers
87
views
When is variance of sample maximum greater than unconditional variance?
Let $X_1$,...,$X_n$ be $n$ i.i.d. RVs with continuous distribution $F$. Further let $X_{(1)}$,...,$X_{(n)}$ be the associated order statistics such that $X_{(1)}<X_{(2)}<...<X_{(n)}$.
Under ...
2
votes
0
answers
130
views
Existence of estimator that reaches Cramer-Rao bound
There is a well known classical result called Cramer-Rao bound:
https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound
Particularly, it is a lower bound for a variance of any unbiased estimate. ...
- 121
2
votes
0
answers
3k
views
variance of multiple variables
Mean or $E(X)$ is linear, so it's valid to write $$E(x_1 + x_2 + x_3) = E(x_1) + E(x_2) + E(x_3)$$ But $Var(x)$ is not linear, so we write $$Var(ax_1 + bx_2 ) = a^2Var(x_1) + b^2Var(x_2) + 2ab\;Cov(...
- 223
2
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0
answers
204
views
Variance of Distributions from the Exponential Family
I want to understand how the variance of an exponential family behaves. To take a very concrete example. Let consider the unit ball $B$ in d dimensions.
Consider the following distribution over unit ...
- 301
2
votes
2
answers
312
views
Analytical expression for variance as a function of the mean value
I have a research problem that seems analogous to a 'draw balls from a bin' problem.
Imagine an experiment where $N$ balls are drawn from an infinite bin containing '0' balls and '1' balls, where $N$ ...
- 31
1
vote
3
answers
740
views
How to measure good or bad luck in roulette
I want to analyze and represent the performance of a bet (X numbers out of 37 total roulette numbers) for a series of spins (N spins).
For example, let's say that I choose 5 numbers (my bet) and these ...
- 123
1
vote
1
answer
473
views
Is the following statement for variance true?
I know: Let be $X$ a random variable and $c\in\mathbb{R}$. Then is
$$Var(cX)=c^2Var(X).$$
But is it true that
$$Var(cX)=\underbrace{Var[Var[\ldots Var}_{c \text{ times}}[X]\ldots]]?$$
- 121
1
vote
2
answers
3k
views
Variance of Random Matrix
Let's consider independent random vectors $\hat{\boldsymbol\theta}_i$, $i = 1, \dots, m$, which are all unbiased for $\boldsymbol\theta$ and that
$$\mathbb{E}\left[\left(\hat{\boldsymbol\theta}_i -
...
- 5,147
1
vote
1
answer
225
views
Variance and covariance inequality
Given a real-valued random variable $X$, is
$$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$
true?
Any pointers for how to tackle this problem would be immensely helpful.
- 549
1
vote
1
answer
91
views
From where term $\left(\frac{1}{n}+\frac{1}{m}\right)$ came in estimated variance of $\bar x - \bar y$
I encountered such a formula for pooled variance:
$$\frac{(n-1)s_x^2+(m-1)s_y^2}{n+m-2}\left(\frac{1}{n} + \frac{1}{m}\right)$$
Here we have two samples of the following sizes $n$ and $m$. $s_x, s_y$...
- 138
1
vote
1
answer
87
views
Confusing variance question
$E(x) = 4$ and $V(x) = 6$
What is the variance of $y=5x+2$?
$E(5x+2) = 5\times 4+2 = 22$
I don't get how the answer is 150
I thought it was $(x-\bar{x})^2$
$(4-22)^2 =324$
very confused
- 177
1
vote
1
answer
728
views
How to represent skewness(X) in terms of the expected value?
Let $X$ be the random variable. $E(X)$ is the expected value of $X$
Then
$Var(X)$ = $E(X^2)$ − $[E(X)]^2$
where $Var(X)$ is the variance of $X$
Then how to represent skewness(X) in terms of the ...
- 113
1
vote
3
answers
1k
views
Calculate variance the right way with two random variables
I'm currently assigning a introductory stats class, and I just can't seem to find out when to use the different variance identities. I have provided an example of an assignment where I got it wrong, ...
- 13
1
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1
answer
77
views
Finding variance?
$\newcommand{\Var}{\mathrm{Var}}$
Consider $Z_i$ as a binary random variable with $\mathrm{Pr}[Z_i = 1] = \pi$. Also, consider $Y_i$ as:
$Y_i|Z_i = 0 \sim \mathrm{Poisson} (\lambda_0) $
$Y_i|Z_i = 1 ...
- 159
1
vote
1
answer
43
views
Unbiased Variance MLE Distribution
If you take $10000$ samples from a normal distribution, the unbiased variance MLE (with Bessel's correction) is
$$\hat{\sigma}^2 = \frac{1}{9999}\sum_i (x_i - \hat{\mu})$$
Apparently the distribution ...
- 503
1
vote
1
answer
64
views
law of total variance derivation without using variance short-cut formula
In the derivation of the law of total variance from the original variance definition (not using the variance short-cut formula), you add and subtract the term $E(y|x)$, group them to the two terms in ...
- 13
1
vote
1
answer
564
views
Variance of expected value, is the formula right?
In this video and this video, I am seeing the variances of expected values calculated as this:
and this:
From which, I derived the formula:
$$\displaystyle\textrm{var}\big(\mathbf E[X\mid Y] \big) =...
- 233
1
vote
1
answer
2k
views
Variance Estimation of MA(1) with known autocovariance function
I haven't worked with time-series in a while now and stumbled upon them in a different setting.
Given $X_t\sim\mathcal{N}(0,\sigma^2)$ for $t=1,\ldots,n$ and the process $Y_t$ for $t=1,\ldots,n-1$ ...
- 13
1
vote
1
answer
153
views
What is $p_i(x_i - \sum_i p_ix_i)$ called?
I was reading a paper (dont have it with me!) on statistics and found a term that I have never encountered before:
$p_{i}{(x_{i}-\mu )} = p_i(x_i - \sum_i p_ix_i)$
After some research it seems that ...
- 1,630
1
vote
1
answer
126
views
Probability - expected value and variance
"A man is playing versus a machine in the following way: The machine chooses 2 numbers randomly from the set of numbers 1,2,3,4,5, where a number can be chosen twice (with replacement). If the ...
- 11
1
vote
1
answer
52
views
Does probability calibration descrease model prediction variance?
Does probability calibration decrease model prediction variance?
Example:
Let's say we have a classifier that is a mail spam detector. It outputs a score between 0-1 to quantify how likely a given ...
- 485
1
vote
1
answer
64
views
Name of the following minimization $E[(X - c)^2] = Var(X) + (E[X] - c)^2$ with $c = E[X]$
My professor proposed the below relationship as a property of the variance (he called $E[(X - c)^2]$ mean squared error):
$$
E[(X - c)^2] = Var(X) + (E[X] - c)^2
$$
and he said that, when $c = E[X]$, ...
- 221
1
vote
1
answer
63
views
Understanding covariance
I came across following problem:
A discrete random variable $P$ takes values $-3,-2,0,2,3$ with probability $0.2$. Let $Q=P^2$ be another random variable. What is covariance of $P$ and $Q$?
I solved ...
- 225
1
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2
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71
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Given $Z\perp X\mid Y$, is it true in general $Var(Z|h(X,Y))=Var(Z|h(c,Y))?$
Given random variables $X, Y, Z$:
If $Z\perp X\mid Y$, then I know that $Var(Z|X,Y)=Var(Z|Y)$
But is it still true in general that
$$Var(Z|h(X,Y))=Var(Z|h(c,Y))?$$
here $h$ is a real valued function ...
- 123
1
vote
2
answers
360
views
Changing only one point of a discrete distribution to maximize variance augmentation
X has a discrete distribution with support $x1, x2, ...$ in $ {]}0,1{[}$. You have the right to change only one of the $xi$ to lead to the highest increase in variance (or, at least, a systematic ...
- 21
1
vote
2
answers
545
views
Two distributions, same mean, different variance: Stochastic dominance for deviation from mean?
Say you have two (bounded) random variables, $X$ and $Y$, on the same discrete probability space, such that $E(X)=E(Y)$ but $Var(X) < Var(Y)$. Do I know that, for any $k \geq 0$,
$$
\text{Prob}(|X-...
- 33
1
vote
1
answer
7k
views
How do I calculate the standard error of the $\chi^2$ statistic?
Question: Suppose that you are testing the idd-ness of a random number generator, and you've done so with the permutation test and the monkey test. Both tests produce a $\chi^2$ statistic and a ...
- 235
1
vote
1
answer
184
views
$Var(Y)$ where $Y = \frac{1}{X}$
Is it possible to estimate variance of $Y$ if we don't know PMF of $X$? More over does it exist? To make question clear, let's assume some array of time stamps differences $ts_{diff} = ts(n) - ts(n-1)$...
- 104
1
vote
1
answer
56
views
If $X=A-F/3$, how to calculate $E(X)$, $Var(X)$ and $P(X≥5)$?
The exercise
An examination of questions with multiple answers, has 20 questions, and each question consists of 4 alternatives, one of which is correct.
The student's score is a random variable $ X $...
- 169
1
vote
1
answer
2k
views
Why is $E(u^2)=Var(y)$? (Binary Response Model)
I'm trying to show some results in binary response models, and a couple of the proofs use the "fact" that $E(u^2)=Var(y)$, but I can't see why this is.
The setup is that $y$ takes on the value $0$ or ...
- 97
1
vote
1
answer
353
views
Variance of function of sample mean
Let's say $X_1, X_2, ..., X_n$ are iid $N(\mu,1)$. We can estimate $\mu$ with $\overline{X}_n$, which is distributed as $N(\mu, 1/n)$. We can estimate $P(X \leq k)$ with $\widehat{\theta}_n = \Phi(k-\...
- 379
1
vote
1
answer
57
views
Variance of $X + \alpha^\top Y$ where $X$ is a scalar random variable and $Y$ is a random vector [duplicate]
Consider a scalar random variable $X\in\mathbb{R}$, a vector random variable $Y\in\mathbb{R}^n$ and a constant (non-random) vector $\alpha\in\mathbb{R}^n$. I want to compute
$$
\mathbb{V}[X + \alpha^\...
- 667
1
vote
1
answer
110
views
Particle detecting Poisson process
Problem:
We are measuring cosmic ray muons.
If we add a lead shielding over the detector, the rate decreases, $\lambda_1=0.1746s^{-1}$
We find the original detection rate is $\lambda_2=0.18 s^{-1}$ (...
- 13
1
vote
1
answer
198
views
If $(X, Y)$ are jointly independent of $Z$, is $Var(X|Y, Z) = Var(X|Y)$?
If $(X, Y)$ are jointly independent of $Z$, then $P(X|Y, Z) = \frac{P(X, Y)P(Z)}{P(Y)P(Z)} = P(X|Y)$.
In this case, is $Var(X|Y, Z) = Var(X|Y)$?
- 2,683
1
vote
1
answer
159
views
$var(y)=b^2var(x)+var(e)$
Suposse $y=xb+e$ where $y$ and $x$ are random variable. $e$ is the error of the regression.
Since x and e are independent then:
$var(y)=b^2var(x)+var(e)$
How can I proof the following double ...
- 344
1
vote
1
answer
41
views
Question relating to joint PDFs
Here are my questions:
Let $X$ ~ Unif$(0, 1)$, and $0<a<b<1$. Also, let
\begin{cases}
Y = 1 & \text{if $0<X<b$} \\
...
- 25