Ams 310 Survey Of Probability And Statistics Worksheet - Midterm Exam I - Stony Brook University Page 11
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11Bonus problem (20 points)
(9). Prove:
1
n
n
p (1
p)
+
p
(1
p) = 1
k
k
=0
=0
Consider the second term in the LHS:
Proof:
( +1)
1
n
n
p
(1
p) =
p
(1
p) .
k
n
k
=0
=0
For each term, change the index
, we have
k = n
k
+1
n
n
n
p (1 p)
=
p (1 p)
=
p (1 p)
.
k
k
k
=
= +1
= +1
Substitute back to the original LHS, we have
n
n
LHS =
p (1
p)
+
p (1
p)
k
k
=0
= +1
n
=
p (1
p)
. = (p + (1
p)) = 1 = 1
k
=0
11
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