Conditional Distribution, Expectation, and Variance

This chapter focuses on conditional distributions and conditional expectation, introduces conditional density functions, and presents problems with both discrete and continuous components. For continuous variables, the conditional (probability) density function plays the analogous role to the conditional probability mass function. A new type of integral is used for both continuous and discrete random variables, and for random variables that exhibit properties of both. A conditional expectation is an expectation computed with respect to a conditional distribution. The conditional variance is derived in a way similar to that of the conditional expectation. The bivariate normal distribution has many remarkable properties, including the fact that both marginal and conditional distributions are normal. The properties and results for the bivariate standard normal distribution extend to the general bivariate normal distribution.