Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, 3rd Edition

12

Estimation of a Random Variable

The techniques in Chapters 10 and 11 use the outcomes of experiments to make inferences about probability models. In this chapter we use observations to calculate an approximate value of a sample value of a random variable that has not been observed. The random variable of interest may be unavailable because it is impractical to measure (for example, the temperature of the sun), or because it is obscured by distortion (a signal corrupted by noise), or because it is not available soon enough. We refer to the estimation of future observations as prediction. A predictor uses random variables observed in early subexperiments to estimate a random variable produced by a later subexperiment. If X is the random variable to be estimated, we adopt the notation (also a random variable) for the estimate. In most of the chapter, we use the mean square error

as a measure of the quality of the estimate.

Signal estimation is a big subject. To introduce it in one chapter, we confine our attention to the following problems:

•  Blind estimation of a random variable

•  Estimation of a random variable given an event

•  Estimation of a random variable given one other random variable

•  Linear estimation of a random variable given a random vector

•  Linear estimation ...