# Search results

1. ### Ampere-Maxwell law

The Maxwell's equations, are differential equations and no laws in the true sense. People just don't seem to get that!? On the other hand, the Lorentz force law, the Coulomb's law, the Biot-Savart's law, the Faraday's law of induction as well as the conservation of charge are the fundamental...
2. ### Deriving the chi-square density function

Homework Statement Consider a standard i.i.d. Gaussian random vector \mathbf{X} = [X_1 \cdot \cdot \cdot X_n]^T and its squared magnitude ||\mathbf{X}||^2 = \sum_{i = 1}^nX_i^2. According to my textbook, to derive the density function of a chi-square random variable ||\mathbf{X}||^2 , one...
3. ### Deriving of the chi-square density function

According to my textbook, to derive the chi-square density function, one should perform three steps. First we consider a standard i.i.d. Gaussian random vector \mathbf{X} = [X_1 \cdot \cdot \cdot X_n]^T and its squared magnitude ||\mathbf{X}||^2 = \sum_{i = 1}^nX_i^2. 1. For n = 1, show that...
4. ### How to derive the multivariate normal distribution

Ok, so now it works out. \mathbf{\Sigma} = \mathbf{A}\mathbf{A}^T is the covariance matrix. Thank you for your help!
5. ### How to derive the multivariate normal distribution

I was on that track before, make use of the CDF and then differentiate back to get the PDF. This is how far I get: Let Y be a standard i.i.d. Gaussian vector. Then use the transformation \mathbf{X} = \mathbf{A}\mathbf{Y} + \mathbf{\mu} P(\mathbf{X} < \mathbf{x}) =...
6. ### How to derive the multivariate normal distribution

If the covariance matrix \mathbf{\Sigma} of the multivariate normal distribution is invertible one can derive the density function: f(x_1,...,x_n) = f(\mathbf{x}) =...