sm8_10

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Problem 8.10 Determine the mean and variance of the sum of five independent uniformly-distributed 
random variables on the interval from -1 to +1. 
 
Solution 
 
Let Xi be the individual uniformly distributed random variables for i = 1,..,5, and let Y be 
the random variable representing the sum: 
 
∑
=
=
5
1i
iXY 
 
Since Xi has zero mean and Var(Xi) = 1/3 (see Problem 8.5), we have 
 
[ ] [ ] 05
1
== ∑
=i
iXY EE 
and 
 
 
 
 
 
 
 
 
( )[ ] [ ]
( )[ ]
[ ] [ ]∑∑
∑
≠=
+=
=
=−=
ji
ji
i
i
i
Y
XXX
X
YYY
EE
E
EE
5
1
2
2
22)(Var µ
 
Since the Xi are independent, we may write this as 
 ( ) [ ] [ ]
3
5
03
5
3
15)(Var
=
+=
+= ∑ ji XXY EE
 
 
 
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