Stat 5101 Lecture Notes - School of Statistics

Chapter 1Random Variables andChange of Variables1.1 Random Variables1.1.1 VariablesBefore we tackle random variables, it is best to be sure we are clear aboutthe notion of a mathematical variable. A variable is a symbol that stands foran unspecified mathematical object, like x in the expression x 2 +2x+1.Often, it is clear from the context what kind of object the variable stands for.In this example, x can be any real number. But not all variables are numerical.We will also use vector variables and variables taking values in arbitrary sets.Thus, when being fussy, we specify the kind of mathematical objects a variablecan symbolize. We do this by specifying the set of objects which are possiblevalues of the variable. For example, we writex 2 +2x+1=(x+1) 2 , x ∈ R,to show that the equality holds for any real number x, the symbol R indicatingthe set of all real numbers.1.1.2 FunctionsIn elementary mathematics, through first year calculus, textbooks, teachers,and students are often a bit vague about the notion of a function, not distinguishingbetween a function, the value of a function, the graph of a function,or an expression defining a function. In higher mathematics, we are sometimesjust as vague when it is clear from the context what is meant, but when clarityis needed, especially in formal definitions, we are careful to distinguish betweenthese concepts.1