30201 06 note that pfalse alarm is rather large for

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: aw of total probability: E [X ] = E [X |A]P (A) + E [X |Ac ]P (Ac ) np nq n(p + q ) n = + = =. 2 2 2 2 2 ] − (E [X ])2 , and apply the law of total To calculate Var(X ) we will use the fact Var(X ) = E [X probability: E [X 2 ] = E [X 2 |A]P (A) + E [X 2 |Ac ]P (Ac ). Here E [X 2 |A] is the second moment of the binomial distribution with parameters n and p, which is equal to the mean squared plus the variance: E [X 2 |A] = (np)2 + npq. Similarly, E [X 2 |Ac ] = 2.10. THE LAW OF TOTAL PROBABILITY, AND BAYES FORMULA 53 (nq )2 + nqp. Therefore, E [X 2 ] = n2 (p2 + q 2 ) (np)2 + npq (nq )2 + nqp + = + npq, 2 2 2 so Var(X ) = E [X 2 ] − E [X ]2 p2 + q 2 1 = n2 − 2 4 = n(1 − 2p) 2 Note that + npq 2 + np(1 − p). (1 − 2p)n , 2 which for fixed p grows linearly with n. In comparison, the standard deviation for a binomial √ random variable with parameters n and p is np(1 − p, which is proportional to n. σX = Var(X ) ≥ 54 2.11 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES Binary hypothesis testing with discrete-type observations The bas...
View Full Document

Ask a homework question - tutors are online