All Questions
17 questions
28
votes
3
answers
4k
views
Confidence Interval for variance given one observation
This is a problem from the "7th Kolmogorov Student Olympiad in Probability Theory":
Given one observation $X$ from a $\operatorname{Normal}(\mu,\sigma^2)$ distribution with both parameters unknown, ...
- 4,329
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 ...
14
votes
2
answers
6k
views
Variance of maximum of Gaussian random variables
Given random variables $X_1,X_2, \cdots, X_n$ sampled iid from $\sim \mathcal{N}(0, \sigma^2)$, define
$$Z = \max_{i \in \{1,2,\cdots, n \}} X_i$$
We have that $\mathbb{E}[Z] \le \sigma \sqrt{2 \log ...
- 689
7
votes
1
answer
716
views
Mean and variance of $\tan(\mathcal{N}(\mu,\,\sigma^{2}))$
How could we find the mean and variance of $\tan(\theta )$ if $\theta \sim \mathcal{N}(\mu,\,\sigma^{2})$?
- 303
5
votes
1
answer
3k
views
How do we derive that $S^2$ is chi-squared distributed (with $n-1$ df)?
The claim is that $$(n-1)S^2/\sigma^2$$ is chi squared distributed with degrees of freedom $n-1$.
$(n-1)S^2/\sigma^2$ can be written as $$\sum_i^n \left(\frac {x_i-\mu}{\sigma}\right)^2-\left(\frac {...
- 2,987
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
4
votes
2
answers
279
views
Variance of random variables involving two independent standard Normals
Let $X$ and $Y$ be two independent standard Normal variables. Let $M := \max(X, Y)$ and $L := \min(X, Y)$. It is given that the covariance between $M$ and $L$ is given by $\text{Cov}(M, L) = 1 / \pi$ ...
- 1,279
3
votes
1
answer
6k
views
What is the probability that a girl is taller than a boy, with boys ~N(68, 4.5) and girls ~N(62, 3.2)?
"Given that boys' heights are distributed normally $\mathcal{N}(68$ inches, $4.5$ inches$)$ and girls are distributed $\mathcal{N}(62$ inches, $3.2$ inches$)$, what is the probability that a girl ...
- 143
2
votes
2
answers
911
views
MLE of variance is biased in a Gaussian distribution
Referring to: How to understand that MLE of variance is biased in a Gaussian distribution
at some point during calculation the formula of the sum of the expected value becomes a single expected value:...
- 33
1
vote
1
answer
4k
views
Does the peak of a Normal Distribution mean anything? [closed]
What does the peak of a Normal distribution show? Let's say if I have a flat peak, does this mean I have a larger variance? What if I have a sharp peak?
For example,
Does the "blue distribution" ...
- 581
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
1
answer
275
views
How to estimate the mean and variance of a Gaussian distribution variable? [closed]
I have two variables 2X and 0.5Y, both are independent and follows Gaussian distribution. How to estimate their mean and variance analytically? I want to know their individual mean and variance, then ...
0
votes
1
answer
133
views
Variance of linear combination of Normal distributions
A company that develops software received an order for a service to be performed within a week and, in order to decide on the profile of the team of programmers to be used, it should take into account ...
- 107
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
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
0
votes
0
answers
47
views
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-...
0
votes
1
answer
117
views
Summation of two Gaussian distributed data with different coefficient of mean and variance
I need some help on Gaussian distribution. i have two dataset, both are identical and independent distributed, but having mean as 2μ_1 and μ_2, same scenario for the variance. How can I add them?
...