Linked Questions

5 votes
1 answer
2k views

Is there a PDF for a generalized non-central chi-squared distribution [duplicate]

Is there a PDF for a distribution defined as a sum of squares of random variables pulled from a family of normal distributions with different standard deviation? Is there a way of analytically ...
0 votes
0 answers
80 views

non-central Chi-square distribution [duplicate]

If $X \sim \mathcal{N}(\mu,\sigma^2)$ where $\mu > 0$ and $\sigma > 2$, how do I find the PDF of $X^2$, and will it correspond to non-central Chi-square distribution?
2 votes
0 answers
61 views

What is the distribution of $(X−Y)^2+(Z−Y)^2$, where $X$,$Y$ and $Z$ are independent normal distributions with their own means and variance? [duplicate]

I came up with a question: What is the distribution of $(X−Y)^2+(Z−Y)^2$, where $X$,$Y$ and $Z$ are independent normal distributions with their own means and variance? The common part is $Y$ in both ...
1 vote
0 answers
41 views

Density Function of $T = (\bar{x})^2$ where $\bar{x}$ $\sim$ $N(\mu, 1/n)$ [duplicate]

Suppose we have $\bar{x}$ $\sim$ $N(\mu, 1/n)$, its pdf then is $f(\bar{x}) = \sqrt{\frac{n}{2\pi}}e^{-n(\bar{x}-\mu)^2/2}$ Let $T = (\bar{x})^2$, we want to calculate its pdf. My first thought is: ...
16 votes
1 answer
19k views

Distribution of the square of a non-standard normal random variable

What is the distribution of the square of a non-standard normal random variable (i.e., the mean is not equal to 0 and the variance is not equal to 1)?
  • 1,031
10 votes
2 answers
5k views

What is the distribution of the sum of squared chi-square random variables?

What would be the distribution of the following equation: $$y = a^2 + 2ad + d^2$$ where $a$ and $d$ are independent non-central chi-square random variables with $2 \textbf{M}$ degrees of freedom. ...
11 votes
1 answer
4k views

What is the expected norm $\mathbb E \lVert X \rVert$ for a multivariate normal $X \sim \mathcal N(\mu, \Sigma)$? [duplicate]

$\DeclareMathOperator\E{\mathbb E} \DeclareMathOperator\Var{\mathrm{Var}} \newcommand\R{\mathbb R} \DeclareMathOperator\N{\mathcal N} \DeclareMathOperator\tr{\mathrm{tr}}$Suppose $X \sim \N(\mu, \...
  • 22.7k
5 votes
3 answers
363 views

What is the PDF of $[(X-a)^2 + (Y-b)^2]^{1/2}$ where $X$ and $Y$ are two non-standard normal random variables?

I have to conduct an experiment getting data from a system. These data are the estimated values, provided by the system, of a true value that we know beforehand. I then compare the estimated values ...
3 votes
0 answers
2k views

Distribution of quadratic form of multivariate normal with linear term

Suppose that $A$ is a symmetric non-random matrix and $X\sim N(\mu,\Sigma)$ and $b \in R^n$ is a non-random vector. Then what is the distribution of $$X^tAX+b^tX \quad ?$$ The distribution without ...
  • 317
3 votes
1 answer
593 views

Sum of squares of non-standard Gaussians

I am looking for the derivation of the density and CDF of the sum $ Y=\sum_{i=1}^{N}(X_i)^2 $ where $X_i \sim \mathcal{N}(0,\sigma^2)$. This problem has been addressed in this question and in this ...
1 vote
1 answer
475 views

Distribution sum of correlated normal variables squared

I'm trying to deduce which distribution my data follows and how to estimate the parameters. I have four random variables $X_i \sim N(\mu_i,\sigma_i^2)$ where the means and variances are all different. ...
  • 13
1 vote
1 answer
353 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X) = \exp(-\lambda ||\...
  • 23
0 votes
0 answers
284 views

Sum of non-central chi-square using Laguerre expansion

I moved this question from my answer on this link. From Castano-Martinez and Lopez-Blazquez(2005), They explained pdf of sum of noncentral Chi-square on equation 3.2, that is: $f(y)= \frac{e^{-\frac{...
  • 13
1 vote
1 answer
113 views

Generating Priors on Lambda for a non-central Chi Distribution of Euclidean Norm of a vector based on component normally distributed elements

I am trying to calculate a posterior predictive distribution for the magnitude (Euclidean norm) of a 3D displacement vector. Displacement in each dimension is independent and normally distributed (but ...
  • 13
0 votes
0 answers
89 views

What is the distribution of cross-sectional volatility?

Assume we have a set of $N$ random variables with known multivariate distribution $\boldsymbol{X}\sim\mathcal{N}(\boldsymbol{\mu},\boldsymbol{\Sigma})$, and a series of realisations $\{\boldsymbol{X_t}...

15 30 50 per page