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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.
 Spring '08
 Zahrn
 The Land

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