Theory of Statistics - George Mason University

1.1 Some Important Probability Facts 11
Random Variables and Probability Distributions
Notice that a random variable is defined in terms only of a measurable space
(Ω, F) and a measurable space defined on the reals (X, B d ). No associated
probability measure is necessary for the definition, but for meaningful applications
of a random variable, we need some probability measure. For a random
variable X defined on (Ω, F) in the probability space (Ω, F, P), the probability
measure of X is P ◦ X −1 . (This is a pushforward measure; see page 706. In
Exercise 1.9, you are asked to show that it is a probability measure.)
A probability space is also called a population, a probability distribution,
a distribution, or a law. The probability measure itself is the final component
that makes a measurable space a probability space, so we associate the distribution
most closely with the measure. Thus, “P” may be used to denote both
a population and the associated probability measure. We use this notational
convention throughout this book.
For a given random variable X, a probability distribution determines
Pr(X ∈ B) for B ⊆ X. The underlying probability measure P of course
determines Pr(X −1 ∈ A) for A ∈ F.
Quantiles
Because the values of random variables are real, we can define various special
values that would have no meaning in an abstract sample space. As we develop
more structure on a probability space characterized by a random variable, we
will define a number of special values relating to the random variable. Without
any further structure, at this point we can define a useful value of a random
variable that just relies on the ordering of the real numbers.
For the random variable X ∈ IR and given π ∈]0, 1[, the quantity xπ
defined as
xπ = inf{x, s.t. Pr(X ≤ x) ≥ π} (1.6)
is called the π quantile of X.
For the random variable X ∈ IR d , there are two ways we can interpret
the quantiles. If the probability associated with the quantile, π, is a scalar,
then the quantile is a level curve or contour in X ∈ IR d . Such a quantile is
obviously much more complicated, and hence, less useful, than a quantile in a
univariate distribution. If π is a d-vector, then the definition in equation (1.6)
applies to each element of X and the quantile is a point in IR d .
Multiple Random Variables on the Same Probability Space
If two random variables X and Y have the same distribution, we write
Theory of Statistics c○2000–2013 James E. Gentle
X d = Y. (1.7)