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:
“It is vain to expect virtue from women till they are, in some degree, independent of men ... Whilst they are absolutely dependent on their husbands they will be cunning, mean, and selfish, and the men who can be gratified by the fawning fondness of spaniel-like affection, have not much delicacy, for love is not to be bought, in any sense of the words, its silken wings are instantly shrivelled up when any thing beside a return in kind is sought.”
—Mary Wollstonecraft (1759–1797)
“The variables of quantification, ‘something,’ ‘nothing,’ ‘everything,’ range over our whole ontology, whatever it may be; and we are convicted of a particular ontological presupposition if, and only if, the alleged presuppositum has to be reckoned among the entities over which our variables range in order to render one of our affirmations true.”
—Willard Van Orman Quine (b. 1908)
“Indeed, I thought, slipping the silver into my purse ... what a change of temper a fixed income will bring about. No force in the world can take from me my five hundred pounds. Food, house and clothing are mine for ever. Therefore not merely do effort and labour cease, but also hatred and bitterness. I need not hate any man; he cannot hurt me. I need not flatter any man; he has nothing to give me.”
—Virginia Woolf (1882–1941)