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(BQ) Approximately 1/3 of the text is new material - this material maintains the style and spirit of previous editions. In order to bridge the gap between concepts and applications, a number of additional examples have been added for further clarity, as well as several new topics. | 370 PROBABILITY AND RANDOM Variables 8-37 Show that in the measurement problem of Example 8-9. he bayesian estimate Ỏ of the parameter 0 equals à - ĩio. J. lgị1 . ơ aồ 9 yvo TX where Ơ X Ĩ -2 ơ0 Ơ II ƠỈ Ơ- II 8-38 Using the ML method find the y confidence interval of the variance V ơ of an A n. Ơ random variable with known mean. 8-39 Show that it fi and 8 are two unbiased minimum variance estimators of a parameter 9. then 81 9-. Hint Form the random variable 9 fl 9 2. Show that or o- l r 2 Ơ . where Ơ is the common variance of 91 and 9 and r is their correlation coefficient. 8-40 The number of successes of an event ri in II trials equals x- . Show that X l - lip. X. - np _ Ắ- - npi Pl lip 1 PiPi where 2 n - X and P A p 1 - p . 8-41 Let Tt.r represent an unbiased estimator for the unknown parameter ip 9 based on the random variables X x . x X under joint density function f x 9 . Show that the Cramer-Rao lower bound for the parameter satisfies the inequality Var Tlx PART II STOCHASTIC PROCESSES 371 CHAPTER 9 GENERAL CONCEPTS 9-1 DEFINITIONS As we recall a random variable X is a rule for assigning to every outcome Ỉ of an experiment 5 a number x c . A stochastic process x t is a rule for assigning to every c a function x r Ị . Thus a stochastic process is a family of time functions depending on the parameter Ị or. equivalently a function of I and Ị. The domain of Í is the set of all experimental outcomes and the domain of t is a set R of real numbers. If R is the real axis then x r is a continuous-time process. If R is the set of integers then x r is a discrete-time process. A discrete-time process is. thus a sequence of random variables. Such a sequence will be denoted by x as in Sec. 7-4 or. to avoid double indices by x n . We shall say that x r is a discrete-stale process if its values are countable. Otherwise it is a continuous-state process. Most results in this investigation will be phrased in terms of continuous-time processes. Topics dealing with discrete-time
