All Questions
Tagged with variance probability
232 questions
4
votes
2
answers
178
views
Finding the variance of a stochastic process
This is part 2 of this question Calculate the mean and variance of a stochastic process?
For the Polya Urn problem, I am trying to understand why the ratio of the variance is:
$$\operatorname{Var}(X_n)...
- 131
2
votes
2
answers
37
views
Predicting the probability distribution of a deterministic dataset
In classical machine learning regression, we often assume the target variable $y$, given an input $x$, follows a probability distribution, allowing us to model and predict not just the expected value ...
- 121
1
vote
1
answer
43
views
Unbiased Variance MLE Distribution
If you take $10000$ samples from a normal distribution, the unbiased variance MLE (with Bessel's correction) is
$$\hat{\sigma}^2 = \frac{1}{9999}\sum_i (x_i - \hat{\mu})$$
Apparently the distribution ...
- 503
0
votes
0
answers
39
views
How to check the Variances between 2 estimators are same or not
Let say I have 2 batches of electric bulb from some manufacturing processes
First batch was run from 10 am to 2 pm (just assume). In this batch total $N_1$ number of bulbs are produced and among them $...
- 405
1
vote
1
answer
48
views
Is the variance of the mean of a set of possibly dependent random variables less than the average of their respective variances? [closed]
Is the variance of the mean of a set of possibly dependent random variables less than or equal to the average of their respective variances?
Mathematically, given random variables $X_1, X_2, ..., X_n$ ...
- 139
1
vote
0
answers
20
views
central moments of random variable from _estimates_ of draws from the distribution function
I am trying to estimate the first two central moments of random variable $r$. The information I have about $r$ is a set of estimates $\hat{r}_i$ for $i \in \mathcal{I}$, each with corresponding ...
0
votes
0
answers
29
views
Expected value of a decreasing function of two random variables
My question is exactly equal to the question posted at Expected value of decreasing function of random variable versus expected value of random variable with just one extra assumption: the two random ...
- 1
0
votes
0
answers
28
views
Constrained Cholesky Decomposition
Suppose that I have an $(n\times 1)$ vector of random variables, $\varepsilon$. However, I know that $k$ linear combinations of $\varepsilon$ are 0. Specifically, I know that for a $(k\times n)$ ...
- 1
0
votes
0
answers
42
views
Conditional Variance of $Z_i|\sum_i\beta_iZ_i$
Let's assume I have $K$ i.i.d. standard normal random variables $Z_1,...,Z_K$. Hence, I know that $V[Z_i] = 1$ and $E[Z_i] = 0$ for all $i\in K$. I am faced with computing the following conditional ...
- 1
0
votes
1
answer
101
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Variance of Multimodal Generalized von Mises Distribution?
How do you calculate the variance of a Multimodal Generalized von Mises (MGvM) distribution? Given its complexity with multiple modes and asymmetry, I'm looking for:
Any formula or method to calculate ...
- 113
3
votes
2
answers
242
views
Calculating $E[(\sum X_i)^4]$
Trying to figure where I'm going wrong with the following. My goal is to calculate var$(\bar X_n^2)$ using $E[(\bar X_n)^4]=\frac{1}{n^4}E[(\sum X_i)^4]$ given that $X_1,...X_n$ are iid with $EX_1=\mu,...
- 385
1
vote
0
answers
86
views
How to find $\mathbb{E} \left[\frac{\bar{\mu}}{\bar{\sigma}^2}\right]$?
I asked the same question on math stacks: MathStacks:, and some user suggest to ask it here for better insight. So this question has found interest in many research problems, but there have been no ...
- 11
0
votes
0
answers
25
views
Question regarding probability and maximum possible variance
I have the following question:
Suppose we have a set of 10 numbers (1, 2, ... , 10), each with a certain probability tagged to it.
Is it true that the highest possible variance is achieved when 1 and ...
- 31
0
votes
0
answers
59
views
mean and variance
Let X be a discrete random variable such that X = 0 with probability 0.5 and X = 1 with probability 0.5. Let Y be a discrete random variable such that Y = 1 when X = 1 and Y = 0 when X = 0. What is ...
- 1
0
votes
0
answers
150
views
What is the distribution of the estimate of residual variance in linear regression? [duplicate]
As the question says, what is the distribution of the estimate of residual variance in a standard gaussian linear regression?
I'm confused because I know in theory the observed $y$ subtract the ...
- 521
5
votes
2
answers
170
views
Bounding the distance of empirical average from its expected value
Suppose we have three sequences of random variables, $(Y_n)_n$, $(W_n)_n$, and $(X_n)_n$ such that:
If $Y_n=a$, then $X_n=b$. If $X_n=b$, then $W_n=c$. That is
$$
1_{[Y_n=a]}\leq 1_{[X_n=b]}\leq 1_{[...
- 935
2
votes
3
answers
396
views
Expected value and variance of median
Suppose $Y|\Lambda\sim U(0,\lambda)$ with $\Lambda \sim U(0,1)$. If there is sample with size $n$ of $Y$ (To simplify, assume $n$ is odd, so $n=2m-1$). How do I calculate the expected value of median (...
- 23
0
votes
0
answers
38
views
What is n when computing the standard error or variance for a statistic computed per 1000?
Let's say we want to calculate the standard error for a statistic that proportion of heads per 1000 coin flips.
So let's say we flip a coin 200 times. We see heads 50 times.
$\hat{\mu}$, our estimate ...
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
$$
\mathbb{V}[X + \alpha^\...
- 667
2
votes
1
answer
86
views
$E[XY]-E[X^2]-E[Y^2]$, is there any special property?
Given probability distributions of random variable $X,Y$, without any additional assumptions, is there any nice representation or properties of the combination $E[XY]-E[X^2]-E[Y^2]$? If not, is there ...
- 23
1
vote
0
answers
26
views
Probability that both the mean and sample variance are both covered by their respective confidence intervals?
I am given the question: "What is the probability that both the mean is in its confidence interval for confidence level a and the variance is in its confidence interval for confidence level a?&...
- 11
0
votes
0
answers
47
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Why Does the Fisher Scoring Algorithm "Work"? [duplicate]
I was reading the following link (https://en.wikipedia.org/wiki/Scoring_algorithm) on the "Fisher Scoring Algorithm". As I understand, the Fisher Scoring Algorithm is similar to the Newton-...
1
vote
0
answers
239
views
Taking derivative of a function containing random variable wrt the variance of that variable [closed]
Say, I have a function containing a random variable such as $ f(X)$, where $X $ is the random variable that comes from a family of random variables that differ only in the first and second moments (e....
- 11
3
votes
1
answer
475
views
What is the conditional $\operatorname{Var}(XY|Y)$ given that $X$ and $Y$ are independent?
What is the conditional $\operatorname{Var}(XY|Y)$ given $X$ and $Y$ are independent?
Is it:
$$\operatorname{Var}(XY|Y)= Y^2\operatorname{Var}(X|Y) = Y^2\operatorname{Var}(X)?$$
- 33
4
votes
3
answers
448
views
How to properly visualize the change of variance of a bivariate Gaussian distribution cut sliced along a fixed variable?
I'm trying to understand the basics of Gaussian Distribution. I struggle to visualice how the variance of the conditional probability of say P (Y|X) changes when X is fixed (given X and Y have a joint ...
- 43
0
votes
1
answer
281
views
Finding mean and variance of number of tosses needed to get exactly 2 heads
A coin with probability of getting head $0.6$ is tossed repeatedly till two heads appear. Let $X$ be the number of tosses needed to get exactly 2 heads. Describe the sample space. Find the mean and ...
- 175
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
0
votes
0
answers
77
views
Maximize Variance of Linear Combination of Matrix Columns
Let $A$ be a $k \times 1$ random vector, and $\mathbf{A}$ be a $n \times k$ matrix of observations.
Letting $t \in \mathbb{R}^{k}$ be a vector of weights s.t. $||t||_2 = 1$, suppose we are interested ...
- 488
3
votes
1
answer
322
views
If X and Y are independent and $E[(XY)^2] = 0$, then $P(X = 0) = 1$ or $P(Y = 0) = 1$
Let $(X, Y)$ be a discrete random vector. Prove that: If $X$ and $Y$ are independent and $E[(XY)^2] = 0$, then $P(X = 0) = 1$ or $P(Y = 0) = 1$
Since $X$ and $Y$ are independent, covariance, i.e., $E(...
- 311
8
votes
2
answers
1k
views
Is variance the area under the curve of the distribution of a population?
I am trying to understand what variance is, I already know the "official" definition
"Variance is the average squared deviations from the mean"
But I am trying to give it a visual ...
- 290
2
votes
1
answer
86
views
When would the variance for a probability distribution give the same result as the standard equation?
Variance equation for a probability distribution:
$$
\sigma^2 = \sum_{i=1}^{N}(x_i-\mu)^2P(X=x_i)
$$
Standard variance equation:
$$
\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2
$$
I understand that ...
- 123
1
vote
0
answers
32
views
Finding variance from normal distribution
Suppose $Z_1$ and $Z2$ ~$N(0,1)$
Let $X_1=2Z_1$ and $X_2=X_1+\frac{\sqrt{3}}{2}Z_2$
Let $Y_1=\sqrt{3}Z_1+Y_2$ and $Y_2=Z_2$
I understand I have to show the mean and variance for $X_1$ and $X_2$ should ...
- 183
0
votes
0
answers
63
views
Variance of $U= a \log (Z+b)-Z$ where $Z$ is the exponential random variable
Consider a random variable
\begin{align}
U= a \log (Z+b)-Z
\end{align}
where $a,b>0$ and $Z$ is an exponential random variable.
Question: Can we find the variance of $U$?
Things that I tried
...
- 205
0
votes
0
answers
33
views
Condition for the asymptotic non-zero point estimation of the variance
we know that a condition for a non-zero point estimate of the variance for a finite sample is that there exist at least two integers $i,j$ such that $X_i\neq X_j$. In other words $\frac{1}{n}\sum\...
1
vote
0
answers
20
views
how do i empirically estimate variance of conditional normal distribution?
I've tried searching for this, but maybe I'm not using the correct search strings. suppose I have joint distribution $P(X_1,X_2)$ over 2 continuous random variables $X_1,X_2$ that I can sample from. ...
- 445
1
vote
1
answer
61
views
If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$
If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$
I have no idea what to address with the conditional expectation part.
Thank you for any comments, someone has ...
- 331
0
votes
0
answers
55
views
How much compensation is needed to take on risk?
We roll three, 8 sided dice. If same face appears 3 times we win 80 dollars. We have a bank of 10,000 dollars. How much are we willing to pay to play? What if we increase the prize to 80,000 dollars? ...
6
votes
1
answer
434
views
Law of Total Variance
I trying to experiment with law of total variance in order to empirically recreate theoretical results.
In particular I am interested in verifying that:
$$
Var(Y) = E[Var(Y|X)] + Var(E[Y|X])
$$
Let's ...
1
vote
1
answer
64
views
Name of the following minimization $E[(X - c)^2] = Var(X) + (E[X] - c)^2$ with $c = E[X]$
My professor proposed the below relationship as a property of the variance (he called $E[(X - c)^2]$ mean squared error):
$$
E[(X - c)^2] = Var(X) + (E[X] - c)^2
$$
and he said that, when $c = E[X]$, ...
- 221
1
vote
1
answer
43
views
Relative part of Variance
The following problem I have $n$ students who are taking a test in which two items of information $X_1$ and $X_2$ are collected. Now I form another variable $X_3=X_1+X_2$ and want to find out how ...
- 13
0
votes
0
answers
160
views
Taking multiple samples from the same population
I am new to stats and I am having difficulty understanding the variances for multiple samples taken from the same population.
Suppose the population weight of a group of men has mean 80 kg and ...
0
votes
0
answers
30
views
When calculating Horwitz-Thompson estimator, is it correct to multiply the pairwise terms of the calculation by two?
I'm currently trying to learn how to calculate the Horwitz - Thompson estimator for population variances. Using this formula
$$ \hat{V}ar(\hat{\tau}_\pi)=\sum\limits_{i=1}^v \left( \dfrac{1-\pi_i}{\pi^...
3
votes
2
answers
390
views
Variance of a random vector (different of the covariance matrix)
Tipically, the variance of a p-dimensional random vector
$$X = (X_1,...,X_p)$$
is defined as a the covariance matrix given by:
$$E[ (X- EX)^T(X- EX) ]$$
But, in the second page of this paper, the ...
- 256
1
vote
0
answers
130
views
Variance of a vector-valued random variable along a unit vector
Let $X$ be a vector-valued random variable with variance $\mathbb{V}[X] < \infty$. How is the variance of $X$ along a unit-vector $\hat{v}$ defined? Can we say that in general it is $\hat{v}^\top \...
- 2,286
0
votes
1
answer
32
views
Standard deviation of discrete variable
A start-up looking to get into the sleeveless shirt market is looking for \$10,000 from investors to get their company started. If you choose to invest this \$10,000, at the end of 5 years the company ...
- 1
14
votes
4
answers
6k
views
Meaning of "Overdispersion" in Statistics
I am trying to understand what "overdispersion" means in statistics.
Based on the Wikipedia page, "overdispersion" is defined as follows : "In statistics, overdispersion is ...
1
vote
0
answers
31
views
Understanding Covariance after Variance (visually)
2 Points i understood from variance derivation-
A) For calculating Variance we do not subtract (or mod add), but rather sum squared all points' differences from the mean.
B) Variance of 1,2,3 will be ...
- 11
1
vote
0
answers
52
views
Correlation Based Models vs Covariance Based Models
I am trying to better understand why some models are "covariance based" vs. why some other models are "correlation based".
1) For example, a Multivariate Normal Distribution ...
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
0
votes
0
answers
178
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Calculating the Standard Deviation of Estimates from a Uniform Distribution
I was looking at this question on Sufficient Statistics and the Uniform Distribution: https://math.stackexchange.com/questions/1359183/why-should-we-care-about-sufficient-statistics
In this question, ...