The Expected Value of a Random Variables

    Suppose we try a thought experiment; flip a coin 500 times, how many heads do you have?

Gambling example

    Definition Let X be a discrete random variable with a probability function p(x). The expected value of X, E(X), is defined as:

    The expected value of a random variable is also called the mean and denoted by m. The probabilty function of a random variable provides a global overview of the variable's behavior. The expected value of a random variable is the most frequently used measure for describing central tendency. This is illustrated below.

• Definitions and Properties

• Expected value

• The Tank Problem

      Do all probability functions have a center?

• St. Petersburg Paradox

    The graph of the histogram above displays only the first 50 probabilities taken on by the random variable. In this case, there is no center.

    Theorem The Law of the Unconscious Statistician

    Variance and Standard Deviation

    In addition to the expected value of a probability function, another useful measure is the variance or spread of the probabilites about the expected value.

    Definition Let X be a discrete random variable with a probability function p(x), and E(X) = m .
     Then the variance of X, written Var(X), is defined as: Var(X) = E[(X - m)²].

Theorem: Var(X) = E(X²) - [E(X)]²
• Variance and Higher Moments

    Moments and Moment Generating Functions

    Let X be a discrete random variable. The moment-generating function for X is denoted M_x(t) and given by: