This preview shows page 1. Sign up to view the full content.
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. DISCRETETYPE RANDOM VARIABLES Binary hypothesis testing with discretetype observations The bas...
View
Full
Document
 Spring '08
 Zahrn
 The Land

Click to edit the document details