How to better understand the multivariate normal distribution?
I am trying to better understand the multivariate normal distribution. Here I try to refer to the conditional distribution part of wiki also the fifth page of this tutorial. I do not quite understand what does the inverse of covariance matrix actually mean.
What is the squared Mahalanobis distance in multivariate normal distribution?
Some things to note about the multivariate normal distribution: This particular quadratic form is also called the squared Mahalanobis distance between the random vector x and the mean vector μ. In this case the multivariate normal density function simplifies to the expression below: Note!
Is the central limit theorem based on multivariate distribution?
Mathematical Simplicity. It turns out that this distribution is relatively easy to work with, so it is easy to obtain multivariate methods based on this particular distribution. Multivariate version of the Central Limit Theorem.
Which is the joint density of a multivariate normal distribution?
If we have a p x 1 random vector X that is distributed according to a multivariate normal distribution with population mean vector μ and population variance-covariance matrix Σ, then this random vector, X, will have the joint density function as shown in the expression below:
Which is the determinant of the variance-covariance matrix?
This covariance is equal to the correlation times the product of the two standard deviations. The determinant of the variance-covariance matrix is simply equal to the product of the variances times 1 minus the squared correlation. The inverse of the variance-covariance matrix takes the form below:
What is the hyper ellipse of a multivariate normal distribution?
Recall the Multivariate Normal Density function below: You will note that this density function, ϕ ( x), only depends on x through the squared Mahalanobis distance: This is the equation for a hyper-ellipse centered at μ. For a bivariate normal, where p = 2 variables, we have an ellipse as shown in the plot below: