# Bivariate Normal Distribution, contour ellipse containing given % samples?

1. Jun 23, 2012

### codiloo

Given a bivariate gaussian distribution,
I'm attempting to find the probability p for which
the ellipse of all points (x,y) for which P(X = x, Y= y) = p contains
a given % of the samples drawn from the distribution.

I want the 2d equivalent for the 1 dimensional case:
given a normal distribution N(0,1):
e.g interval between points with p = 0.24197072 contains 68.2% of all samples
e.g interval between points with p = 0.05399097 contains 95.4% of all samples
e.g interval between points with p = 0.00013383 contains 99.6% of all samples
in two dimensions these interval boundries become an ellipse and I'm interested in finding the p value corresponding to a given % (contained samples in contour ellipse with p) value in the 2 dimensional case.

Some extra info:
A matlab, python (using numpy, scipy?) numerical approximation is ok, I don't need an analytic formula.

Actually I just want to draw the ellipses containing 75%, 95%, 99% of the samples in python (using matlibplot) for a given gaussian distribution (varying mean & covariance). I know how to do this if I obtain p first (contour plots).

Thank you for reading my question and I hope you can help.

2. Jun 23, 2012

### viraltux

Hi codiloo,

The probability hyper-ellipsoid hyper-volume for a multivariate normal follows

$$(x-μ)^T \Sigma^{-1}(x-μ) ≤ χ^2_k(p)$$

Where x is a k-dimensional vector, μ is the k-dimensional mean vector, Ʃ is the variance-covariance matrix and $χ^2_k(p)$ is the p quantile of the chi-square distribution with k degrees of freedom.

When k = 2 dimensions the expression represents the area of the ellipse you are asking for, and $χ^2_2$ behaves as an exponential distribution.

Last edited: Jun 23, 2012
3. Jun 24, 2012

### haruspex

After a huge calculation involving rotating co-ordinates I ended up with
P[(x, y) lies inside the contour pdf(x,y) = k] = 1 - 2πkD, where D is the determinant of the covariance matrix, i.e. = √(σ12σ22 - ρ4).
Note e.g. that the peak pdf value is 1/2πD
If it's right, there must be an easier way.

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