The Expected Value (or expectation, mean) of a random variable can be thought of as the "weighted average" of the possible outcomes of the random variable. Mathematically, if , are all of the possible values that X can take, the expected value of X can be calculated as follows:
Linearity of Expectation
The most important property of expected value is Linearity of Expectation. For any two random variables X and Y, a and b scaling coefficients and c is our constant, the following property of holds:
The above is true regardless of whether X and Y are independent.
Conditional distributions are still distributions. Treating them as a whole and applying the definition of expectation gives:
Variance tells us how spread out the distribution of a random variable is. It is defined as
Properties of Variance
if and are independent
Indicator Random Variables are random variables whose value is 1 when a particular event happens, or 0 when it does not. Let be an indicator random variable for the event . Then, we have:
Suppose . Then, because has a chance of being 1, and a chance of being 0.
Properties of Indicators
, and for any power .
is the indicator for the event (that is, if and only if and occur, and 0 otherwise)
The fundamental bridge is the idea that . When we want to calculate the expected value of a complicated event, sometimes we can break it down into many indicator random variables, and then apply linearity of expectation on that. For example, if , then: