Download EX: Find the probability density function of the average value of 12

CONCEPTUAL TOOLS
By: Neil E. Cotter
P ROBABILITY
LINEAR COMBINATIONS RV'S
Normal/gaussian dist
EXAMPLE 1
EX:
Find the probability density function of the average value of 12 independent standard
gaussian random variables.
SOL'N:
We want to find the probability density function for Y:
12
1
Y = ∑ Xi
12
i=1
where
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Xi ~ n(0,1) are independent standard gaussian random variables
We rewrite the expression for Y to emphasize that it is a linear combination
of independent random variables.
Y=
1
1
X1 + K + X12
12
12
Thus, we have the following mean and variance:
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µY =
1
1
1
µ X1 + K + µ X12 = 12 ⋅ ⋅ 0 = 0
12
12
12
 1 2
 1 2
 1 2
1
σY2 =   σ 2X + K +   σ 2X = 12 ⋅   ⋅1 =
1
12
12 
 12 
12 
12
Also, the probability density function (pdf) for a linear combination of
independent gaussian random variables is a gaussian random variable. Thus,
the pdf for Y is the following gaussian distribution:
fY (y) =
NOTE:
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1
2πσY2
e
−(y−µY ) 2 / 2σ Y2
The pdf for a linear combination of independent gaussian
random variables, Xi, is gaussian even when the Xi have differing
means and variances.