# If 22 2 0132 but mh and mv were independent we would

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Unformatted text preview: ctice using the convolution formula (4.14). Since the support of X is {0, 1, . . . , m} and the support of Y is {0, 1, . . . , n}, the support of S is {0, 1, . . . , m + n}. So select an integer k with 0 ≤ k ≤ m + n. Then (4.14) becomes min(m,k) mj n p (1 − p)m−j pk−j (1 − p)n−k+j j k−j j =0 min(m,k) m n pk (1 − p)m+n−k = j k−j pS (k ) = j =0 = m+n k p (1 − p)m+n−k , k + where the last step can be justiﬁed as follows. Recall that mk n is the number of subsets of size k that can be formed by selecting from among m + n distinct objects. Suppose the ﬁrst m objects are orange and the other n objects are blue. Then the number of subsets of size k that can be made n using the m + n objects, such that j of the objects in the set are orange, is m k−j , because the j j orange objects in the set can be chosen separately from the k − j blue objects. Then summing + over j gives mk n . This veriﬁes the equation, and completes the second derivation of the fact tha...
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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.

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