Random Variable - Functions of Random Variables

Functions of Random Variables

A new random variable Y can be defined by applying a real Borel measurable function to the outcomes of a real-valued random variable X. The cumulative distribution function of is

If function g is invertible, i.e. g−1 exists, and increasing, then the previous relation can be extended to obtain

and, again with the same hypotheses of invertibility of g, assuming also differentiability, we can find the relation between the probability density functions by differentiating both sides with respect to y, in order to obtain

.

If there is no invertibility of g but each y admits at most a countable number of roots (i.e. a finite, or countably infinite, number of xi such that y = g(xi)) then the previous relation between the probability density functions can be generalized with

where xi = gi-1(y). The formulas for densities do not demand g to be increasing.

In the measure-theoretic, axiomatic approach to probability, if we have a random variable on and a Borel measurable function, then will also be a random variable on, since the composition of measurable functions is also measurable. (However, this is not true if is Lebesgue measurable.) The same procedure that allowed one to go from a probability space to can be used to obtain the distribution of .

Read more about this topic:  Random Variable

Famous quotes containing the words functions of, functions, random and/or variables:

    One of the most highly valued functions of used parents these days is to be the villains of their children’s lives, the people the child blames for any shortcomings or disappointments. But if your identity comes from your parents’ failings, then you remain forever a member of the child generation, stuck and unable to move on to an adulthood in which you identify yourself in terms of what you do, not what has been done to you.
    Frank Pittman (20th century)

    The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.
    Cyril Connolly (1903–1974)

    Novels as dull as dishwater, with the grease of random sentiments floating on top.
    Italo Calvino (1923–1985)

    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)