Stat 5101 Lecture Notes - School of Statistics

1.1. RANDOM VARIABLES 5that is, X is the identity function on S.As we mentioned in our previous discussion of identity functions, when you’resloppy in terminology and notation the identity function disappears. If you don’tdistinguish between functions, their values, and their defining expressions x isboth a variable and a function. Here, sloppiness causes the disappearance ofthe distinction between the random variable “big X” and the ordinary variable“little s.” If you don’t distinguish between the function X and its values X(s),then X is s.When we plug in X(s) =sinto the expression (1.6), we getB = { s ∈ S : s ∈ A } = A.Thus when X is the identity random variable P (X ∈ A) is just another notationfor P (A). Caution: when X is not the identity random variable, this isn’t true.Another Useful NotationFor probability models (distributions) having a standard abbreviation, likeExp(λ) for the exponential distribution with parameter λ we use the notationX ∼ Exp(λ)as shorthand for the statement that X is a random variable with this probabilitydistribution. Strictly speaking, X is the identity random variable for the Exp(λ)probability model.ExamplesExample 1.1.1 (Exponential Random Variable).SupposeX ∼ Exp(λ).What isP (X >x),for x>0?The definition of the probability measure associated with a continuous probabilitymodel says∫P (A) = f(x)dx.We only have to figure what event A we want and what density function f.ATo calculate the probability of an event A. Integrate the density overA for a continuous probability model (sum over A for a discretemodel).The event A isA = { s ∈ R : s>x}=(x, ∞),