Chapter List

Independent Random Variables

Let X ₁ , , X _N be N independent real random variables with the same mean (that is, expected value) ? and same variance ? ². The main consequence of independence is that E( X _i X _j) = E( X _i) E( X _j) = ? ² for i ? j. Then, it is easily shown that the sample average

has ? for its mean and ? ² /N for its variance.

Exercise 39.1. Prove these two assertions.

Maximum Likelihood Parameter Estimation

Suppose that the random variable X has a probability density function p( x; ?), where ? is an unknown parameter. A common problem in statistics is to estimate ? from independently sampled values of X, say x ₁ , , x _N. A frequently used approach is to maximize the function of ? given by

The function L( ?) is the likelihood function and a value of ? maximizing L( ?) is a maximum likelihood estimate. We give two examples of maximum likelihood (ML) estimation.