M5L9
Functions of Multiple Random Variables-2
1.
Introduction
In this lecture, various statistical properties of one function of two random variables are
discussed. The sum and difference of independent Normal variates, maximum, minimum
product and quotient of two random variables are discussed in detail along with numerical
examples.
2.
Sum and Difference of Normal Variates
If
and
are independent normal variates, having mean as
X
,
Y
and standard deviation as
X
,
Y
respectively.
Y
X
Z
is one function for these two variables and its pdf can be
expressed as,
dy
y
f
y
z
f
z
f
Y
X
Z
Now this equation for pdf of
Z
can be expanded as,
dy
by
ay
exp
z
exp
dy
y
y
z
exp
z
f
X
X
Y
Y
Y
X
Y
Y
X
X
Y
X
Z
2
2
1
2
1
2
1
2
1
2
1
2
2
2
2
2
where,
2
2
2
2
1
1
Y
Y
X
X
Y
X
z
b
and
a
Now, considering the last integral part of the expression for pdf and taking a substitution
, we get,
a
b
exp
a
dw
aw
exp
e
dy
by
ay
exp
a
/
b
2
2
2
1
2
2
1
2
2
2
2
2
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