def. Random Variable A Random Variable from probability space to measurable space is a function:

  • is the Borel sigma-algebra on real numbers (=Borel set)
    • This is a set of open intervals on that satisfies -algebra properties
  • for all , it is true that
    • You’re given an open interval (=set) on real numbers
    • Get all such that is within this range (=pre-image)
    • This set (a subset of ) must be in the sigma-algebra of
    • This applies to any range
    • & We can then say ” is -measurable” or ” has enough information to measure “.
    • Intuition. Let there be other sigma-algebras such that , in addition to being -measurable. In this case…
      • is -measurable because has more information than .
      • However, is not -measurable because has less information than .

def. Probability on a Random Variable. Probability function on random variable is a function such that:

  • & i.e., the probability we encounter daily is simply a shorthand notation
  • You put the interval , and get the probability of all in that interval’s pre-image
  • thus by abuse of notation we write:

Example. Let is the trajectory of a coin toss. This is a very big set, and probably also infinite. On the other hand, let a random variable for

    • This is much simpler and more useful.
  • This means .
  • If , then we can try to measure the probability of
  • Because we know the pre-image of is in , we know that is defined.

thm. Addition Rule for Random Variables. For a discrete random variable :

Functions of Random Variables

Motivation. Functions can be made of random variables; for example let . In order to investigate , need a way to derive the probability distribution of from .

Example. Let be a random variable defined by a function of another random variable ; . Then:

thm. Random variables are equal when:

Indicator Functions

def. Indicator Functions. For event , the indicator function is a random variable (i.e. function) such that:


  • Indicator functions are useful in probability for solving problems, not for being a fundamental mathematical object.
  • Remember that indicator functions are also random variables. All the rules for random variables apply, including the identities for expected values.

Properties. let an indicator function describing an event with probability , then: