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# 25&igrave;ž&yen; - Homework 1 25.2 1(a Find a...

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Homework 1 § 25 . 2 1. (a) Find a maximum likelihood estimate for θ = p in the case of the binomial distribution. (b) Extend (a) as follows. Suppose that m times n trials were made and in the ﬁrst n trials A happened k 1 times, in the second trials A happened k 2 times, ··· , in the m th n trials A happened k m times. Find a maximum likelihood estimate of p based on this information. Sol. (a) Let f ( x ) = ± n x p x q n - x and l = f ( x 1 ) f ( x 2 ) ··· f ( x n ) = Mp s (1 - p ) n 2 - s where M = ± n x 1 ¶± n x 2 ··· ± n x n and s = x 1 + x 2 + ··· + x n . ln l = ln M + s ln p + ( n 2 - s )ln(1 - p ) and d ln l dp = s p - ( n 2 - s ) 1 1 - p = 0 and then p = s n 2 = ¯ x n . Thus ˆ p = ¯ x n . (b) Let f ( x ) = ± n x p x q n - x and l = f ( k 1 ) f ( k 2 ) ··· f ( k m ) = Mp s (1 - p ) nm - s where M = ± m k 1 ¶± m k 2 ··· ± m k m and s = k 1 + k 2 + ··· + k m . ln l = ln M + s ln p + ( nm - s )ln(1 - p ) and d ln l dp = s p - ( nm - s ) 1 1 - p = 0 and then p = s nm = ¯ x n . Thus ˆ p = ¯ x n . 2. Consider X = Number of independent trials until an event A occurs . Show that X has the probability function f ( x ) = pq x - 1 , x = 1 , 2 , ··· , where p is the probability of A in a single trial and q = 1 - p . Find the maximum likelihood estimate of p (CSE) C. K. Ko

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Homework 2 corresponding to a sample x 1 , ··· ,x n of observed values of X . Sol.
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## This note was uploaded on 11/08/2011 for the course EE 111 taught by Professor Kim during the Spring '11 term at Korea University.

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25&igrave;ž&yen; - Homework 1 25.2 1(a Find a...

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