Tagged Questions

0
votes
2answers
62 views

Simple homework question about normally distributed variables

The question states: Consider a set of random variables $X_i$, where $i=1,...n$. Each $X_i$ is normally distributed with mean $0$ and variance $1$, i.e. $X_i$ are $\mathcal N(0,1)$. What is ...
0
votes
0answers
113 views

Let X1,X2,…,Xn be i.i.d. N(θ1, θ2), please prove that E[(x1-θ1)^4] = 3θ2^2

If x$_{1}$, x$_{2}$,...,x$_{n}$ is sampled from N($\theta$$_{1}$, $\theta$$_{2}$), how can I prove that E [(x$_{1}$ - $\theta$$_{1}$)$^{4}$] = 3$\theta$$_{2}$$^{2}$? I started off this question ...
1
vote
0answers
130 views

Finding the UMVUE of the variance of a gaussian with mean zero

Given $Z_1, ..., Z_n, \sim\mathcal{N}(0, θ^2), θ>0$. Define $X_i=|Z_i|$ and consider estimation of $\theta$ and $θ^2$ on the basis of the random sample $X_1,...X_n$. Find the uniformly minimum ...
0
votes
1answer
87 views

Spherical Gaussian Sigma dimension

I think I am confused with this thing. If we have a 3 dimension Gaussian then the MLE estimate for $\mu$ is a vector with 3 element $$\mu(1)' = \frac{1}{n}\sum_{j = 1} ^ n x_j\text{ and so on ...
1
vote
1answer
101 views

Distribution which has maximum or minimum variance for a given entropy

We know that the normal distribution has the maximum entropy among all continuous distribution on $\mathbb{R}$ for a given variance. I wonder what's the opposite, i.e. what distribution has the ...
15
votes
3answers
383 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 ...
4
votes
1answer
83 views

Monomial distribution of $X^a \cdot Y^b$

What is the distribution of the following monomial? $$X^a \cdot Y^b$$ where $X$ and $Y$ are normal random variables and $a$ and $b$ are natural numbers. For example, when $X \sim N(0,1)$, $a=2$, and ...
1
vote
1answer
319 views

Are logistic regression coefficient estimates biased when the predictor has large variance?

I'm simulating data from a logistic regression model: log(p/1-p)= 0 + X where $X \sim N(0,\sigma^2)$. After I simulate the data, I fit a logistic regression ...
3
votes
2answers
777 views

Why is the variance of $X-Y$ equal to the sum of the variances when $X,Y$ are independent?

I have one question about this. I know that if we have $\mathrm{X}_1,\mathrm{X}_2,\ldots,\mathrm{X}_n$ independent and normally distributed random variables, then the sum ...
0
votes
0answers
119 views

Is 2-sigma limit applicable as a measure of variance to a distribution?

There is a paper that uses something like the 2-sigma limit to check if a quantity is more-or-less constant. More specifically in this paper on Text Detection (Sec. 3.2 3rd para last line) this limit ...
0
votes
1answer
594 views

Mean of a product of Gaussians

If I have two normally distributed random variables, X and Y, and I want to find the mean of the distribution that results from multiplying them together, which of the following two formulas should I ...
4
votes
2answers
467 views

Is the variance of the multivariate folded normal distribution known?

The mean and variance of the folded normal distribution are known. Consider now the distribution of $(|x_1|, \ldots, |x_n|)$, where $\mathbb{x} \sim N(\mu, \Sigma)$. The mean of the multivariate ...
1
vote
1answer
528 views

How to deal with a non-gaussian distribution & heteroscedasticity

I am working on my thesis project and have come across a problem with the statistics which I am looking for a bit of guidance. I am running ANOVA tests to determine significance between groups but i ...
4
votes
0answers
168 views

How does number of observations supporting alternate hypothesis on a test of a variance have to scale so that null is rejected?

Informal explanation: In the course of my research I've run into the following problem: I am observing a machine that outputs random numbers. Most (if not all) of these random numbers come from the ...
5
votes
3answers
458 views

Is my data distribution normal? (Tried Shapiro and Kolmogornov-Smirnov tests)

I have a 1D data set with 83163 data points, and I would like to know whether the data follows a normal distribution. I tried using shapiro.test and ks.test in R: d is a vector containing the data ...
0
votes
1answer
240 views

Mean and variance of a normally distributed random number created from the average of a set of uniformly distributed random numbers

An old-fashioned way of generating normally distributed random numbers entailed setting each normally distributed random number equal to the average of a set of uniformly distributed random numbers, ...
2
votes
1answer
417 views

What is the variance of the sum of components of a multivariate normal distribution?

I'm talking with my advisor about how to compute standard deviations for, say, combined standardized test scores for admissions purposes. For example, we'd be interested to compute the sum of the ...
6
votes
1answer
228 views

Estimating the variance of poker win rates

Suppose you have a casino with n poker players. Each player has a win rate - the amount of money he wins or loses per hand. We assume that these win rates are normally distributed with a mean of 0. ...
3
votes
2answers
815 views

When estimating variance, why do unbiased estimators divide by n-1 yet maximum likelihood estimates divide by n?

I am totally confused: On the one hand you can read all kinds of explanations why you have to divide by n-1 to get an unbiased estimator for the (unknown) population variance (degrees of freedom, not ...
6
votes
2answers
157 views

Inference with Gaussian Random Variable

Let $X = N(0,\frac{1}{\alpha})$, $Y = 2X + 8 + N_{y}$, and $N_{y}$ be a noise $N_{y} = N(0,1)$. Then, $P(y|x) = \frac{1}{\sqrt{2\pi}}exp\{ -\frac{1}{2}(y - 2x - 8)^{2} \}$ and $P(x) = ...
7
votes
1answer
128 views

How to check for bivariate Gaussianity without the use of regression?

What steps could be taken to check for bivariate Gaussianity without using regression based check? Can we somehow employ the use of definition of variogram measure for assessing spatial variability?
6
votes
3answers
869 views

What is the probability that a normal distribution with infinite variance has a value greater than its mean?

I got asked something similar to this in interview today. The interviewer wanted to know what is the probability that an at-the-money option will end up in-the-money when volatility tends to ...