Lecture 6/ 2 Joint and Conditional Distributions of Two variables/3rd Stage / 2nd Course /2018-2019

: Expectation and Variance In the previous chapter we looked at probability, with three major themes: 1. Conditional probability: P(A | B). 2. First-step analysis for calculating eventual probabilities in a stochastic process. 3. Calculating probabilities for continuous and discrete random variables. In this chapter, we look at the same themes for expectation and variance. The expectation of a random variable is the long-term average of the random variable. Imagine observing many thousands of independent random values from the random variable of interest. Take the average of these random values. The expectation is the value of this average as the sample size tends to infinity. We will repeat the three themes of the previous chapter, but in a different order. 1. Calculating expectations for continuous and discrete random variables. 2. Conditional expectation: the expectation of a random variable X, conditional on the value taken by another random variable Y . If the value of Y affects the value of X (i.e. X and Y are dependent), the conditional expectation of X given the value of Y will be different from the overall expectation of X. 3. First-step analysis for calculating the expected amount of time needed to reach a particular state in a process (e.g. the expected number of shots before we win a game of tennis). We will also study similar themes for variance. 45 3.1 Expectation The mean, expected value, or expectation of a random variable X is written as E(X) or µX. If we observe N random values of X, then the mean of the N values will be approximately equal to E(X) for large N. The expectation is defined differently for continuous and discrete random variables. Definition: Let X be a continuous random variable with p.d.f. fX(x). The expected value of X is E(X) = Z ? ?? xfX(x) dx. Definition: Let X be a discrete random variable with probability function fX(x). The expected value of X is E(X) = X x xfX(x) = X x xP(X = x). ensus in the Voter Process.