Given an event

Let be an event with . Given a discrete Random variable , the conditional expectation , given , is expressed as

If is continuous, the expectation becomes an integral instead.

Here, is called the conditional PDF and is defined as the derivative of the conditional CDF .

Given a random variable

Let and be random variables. Let be a function that takes any real number and produces . Then the conditional expectation of , given , is given by the random variable defined by .

The distinction between what is a constant and what is a random variable is important.

  • remains a constant.
  • is also a constant.
  • is a random variable. It is defined as , which is a transformation of , a random variable (here, we don’t know that for some ).


  • If and are Independent, then .
  • for any function . Intuitively, we are given , so we can treat as a constant and move it out under expectation properties. In other words, and .
  • .