Conditioning (probability) - Conditioning On The Level of Densities

Conditioning On The Level of Densities

Example. A point of the sphere x2 + y2 + z2 = 1 is chosen at random according to the uniform distribution on the sphere. The random variables X, Y, Z are the coordinates of the random point. The joint density of X, Y, Z does not exist (since the sphere is of zero volume), but the joint density f_X,Y of X, Y exists,

$f_{X,Y} (x,y) = \begin{cases} \frac1{2\pi\sqrt{1-x^2-y^2}} &\text{if } x^2+y^2<1,\\ 0 &\text{otherwise}. \end{cases}$

(The density is non-constant because of a non-constant angle between the sphere and the plane.) The density of X may be calculated by integration,

surprisingly, the result does not depend on x in (−1,1),

$f_X(x) = \begin{cases} 0.5 &\text{for } -1<x<1,\\ 0 &\text{otherwise}, \end{cases}$

which means that X is distributed uniformly on (−1,1). The same holds for Y and Z (and in fact, for aX + bY + cZ whenever a2 + b2 + c2 = 1).

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