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Unformatted text preview: ls. The coin shows heads with probability
p each time it is ﬂipped. The student ﬂips the coin n times for some large value of n (for example,
n = 1000 is reasonable). Heads shows on k of the ﬂips. Find the ML estimate of p.
Solution: Let X be the number of times heads shows for n ﬂips of the coin. It has the binomial
distribution with parameters n and p. Therefore, the likelihood that X = k is pX (k ) = n pk (1 −
k
p)n−k . Here n is known (the number of times the coin is ﬂipped) and k is known (the number
of times heads shows). Therefore, the maximum likelihood estimate, pM L , is the value of p that
maximizes n pk (1 − p)n−k . Equivalently, pM L , is the value of p that maximizes pk (1 − p)n−k . First,
k
assume that 1 ≤ k ≤ n − 1. Then
d(pk (1 − p)n−k )
=
dp k n−k
−
p
1−p pk (1 − p)n−k = (k − np)pk−1 (1 − p)n−k−1 . 44 CHAPTER 2. DISCRETETYPE RANDOM VARIABLES k
k
This derivative is positive if p ≤ n and negative if p ≥ n . Therefore, the likelihood is maxi...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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