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:
“The awareness that health is dependent upon habits that we control makes us the first generation in history that to a large extent determines its own destiny.”
—Jimmy Carter (James Earl Carter, Jr.)
“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)
“A man may see how this world goes with no eyes. Look with thine ears;
see how yond justice rails upon yond simple thief. Hark in thine ear: change places, and handy-dandy, which is the justice, which is the thief?”
—William Shakespeare (1564–1616)