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 fX,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).

Read more about this topic:  Conditioning (probability)

Famous quotes containing the words conditioning and/or level:

    “The climacteric marks the end of apologizing. The chrysalis of conditioning has once for all to break and the female woman finally to emerge.”
    —Germaine Greer (b. 1939)

    “... when you do get a job everybody says, “Well, they wanted a black woman,” which necessarily puts you on a level where you have to prove yourself above being a woman and being black.... Now, I would say, in certain situations, it helped me simply because I was mildly attractive, not because I was black or a woman. That gets you more mileage than anything else.... God help you if you’re not an attractive woman.”
    —Theresa Brown (b. 1957)