1 - Random variables

Summary

Random variables

Statistics is about extracting information from data that contain an inherently unpredictable component. Random variables are the mathematical construct used to build models of such variability. A random variable takes a different value, at random, each time it is observed. We cannot say, in advance, exactly what value will be taken, but we can make probability statements about the values likely to occur. That is, we can characterise the distribution of values taken by a random variable. This chapter briefly reviews the technical constructs used for working with random variables, as well as a number of generally useful related results. See De Groot and Schervish (2002) or Grimmett and Stirzaker (2001) for fuller introductions.

Cumulative distribution functions

The cumulative distribution function (c.d.f.) of a random variable (r.v.), X, is the function F(x) such that

F(x) = Pr(X ≤ x).

That is, F(x) gives the probability that the value of X will be less than or equal to x. Obviously, F(−∞) = 0, F(∞) = 1 and F(x) is monotonic. A useful consequence of this definition is that if F is continuous then F(x) has a uniform distribution on [0, 1]: it takes any value between 0 and 1 with equal probability. This is because

Pr(X ≤ x) = Pr{F(x) ≤ F(x)} = F(x) ⇒ Pr{F(x) ≤ u} = u

(if F is continuous), the latter being the c.d.f. of a uniform r.v. on [0, 1].

Define the inverse of the c.d.f. as F⁻ (u) = min(x|F(x) ≥ u), which is just the usual inverse function of F if F is continuous. F⁻ is often called the quantile function of X. If U has a uniform distribution on [0, 1], then F⁻ (U) is distributed as X with c.d.f. F. Given some way of generating uniform random deviates, this provides a method for generating random variables from any distribution with a computable F⁻.