Isye 2027

# For example if x 2 is observed then h1 is declared

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Unformatted text preview: half, and then ﬂip the selected coin n times. Let X be the number of times heads shows. Compute the pmf, mean, and standard deviation of X. Solution: Let A be the event that the ﬁrst coin is selected. By the law of total probability, pX (k ) = P {X = k } = P ({X = k }A) + P ({X = k }Ac ) = P (X = k |A)P (A) + P (X = k |Ac )P (Ac ) 1 n k n−k 1 n k n−k = pq + qp . 2k 2k 52 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES The two conditional pmfs and the unconditional pmf of X are shown in Figure 2.10 for n = 24 and p = 1/3. Figure 2.10: The pmfs of X for n = 24 and p = 1/3. Top left: conditional pmf of X given the ﬁrst coin is selected. Top right: conditional pmf of X given the second coin is selected. Bottom: The unconditional pmf of X. The two plots in the top of the ﬁgure are the binomial pmf for n = 24 and parameter p = 1/3 and the binomial pmf for n = 24 and the parameter p = 2/3. The bottom plot in the ﬁgure shows the resulting pmf of X . The mean of X can be calculated using the l...
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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 Tech.

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