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homework03

# homework03 - Probability for Engineering Applications...

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Probability for Engineering Applications Assignment #3, due at the beginning of class on Thursday, February 9 th , 2006 1. (2 pts) (Textbook exercise 2.47) In this problem we derive the multinomial coeffi- cient. Suppose we partition a set of n distinct objects into m subsets B 1 , B 2 , · · · , B m of size k 1 , k 2 , · · · , k m , respectively, where k i 0 i and k 1 + k 2 + · · · + k m = n . a. Let N i denote the number of possible outcomes when the i th subset is selected. Show that N 1 = n k 1 ! , N 2 = n - k 1 k 2 ! , · · · , N m - 1 = n - k 1 - · · · - k m - 2 k m - 1 ! , N m = 1 Basically we are interested in the following: We have a total on n objects. From these, we first pick k 1 objects. How many ways of doing this (denote this by N 1 )? From the remaining objects, we now pick k 2 objects. How many ways of doing this (denote this by N 2 )? And so on. b. Show that the total number of partitions is then N 1 N 2 · · · N m = n ! k 1 ! k 2 ! · · · k m ! 2. (1 pt) (Textbook exercise 2.49) Show that P [ A | B ] satisfies the axioms of probability: i. 0 P [ A | B ] 1 ii. P [ S

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