Isye 2027

# 30201 06 note that pfalse alarm is rather large for

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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...
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