Probability Density Function - Dependent Variables and Change of Variables

Dependent Variables and Change of Variables

If the probability density function of a random variable X is given as fX(x), it is possible (but often not necessary; see below) to calculate the probability density function of some variable Y = g(X). This is also called a “change of variable” and is in practice used to generate a random variable of arbitrary shape fg(X) = fY using a known (for instance uniform) random number generator.

If the function g is monotonic, then the resulting density function is

Here g−1 denotes the inverse function.

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,

or

For functions which are not monotonic the probability density function for y is

where n(y) is the number of solutions in x for the equation g(x) = y, and g−1k(y) are these solutions.

It is tempting to think that in order to find the expected value E(g(X)) one must first find the probability density fg(X) of the new random variable Y = g(X). However, rather than computing

one may find instead

The values of the two integrals are the same in all cases in which both X and g(X) actually have probability density functions. It is not necessary that g be a one-to-one function. In some cases the latter integral is computed much more easily than the former.

Read more about this topic:  Probability Density Function

Famous quotes containing the words dependent, variables and/or change:

    “For my part, I have no hesitation in saying that although the American woman never leaves her domestic sphere and is in some respects very dependent within it, nowhere does she enjoy a higher station . . . if anyone asks me what I think the chief cause of the extraordinary prosperity and growing power of this nation, I should answer that it is due to the superiority of their woman.”
    —Alexis de Tocqueville (1805–1859)

    “The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.”
    —George Steiner (b. 1929)

    “If we are to change our world view, images have to change. The artist now has a very important job to do. He’s not a little peripheral figure entertaining rich people, he’s really needed.”
    —David Hockney (b. 1937)