Consider a
two-dimensional random vector Z, whose
components are independent and both have U(0, 1)
marginal PDFs. Let Y = Z1 + Z2.
Use moment-generating functions to calculate E(Y2).
Solution
By [3.129], the moment generating
function of Y is:
[s1]
[s2]
[s3]
[s4]
As indicated on p. 144, the second moment E(Y2)
of Y is obtained as the second derivative of MGF [s4] evaluated at
w = 0:
[s5]
[s6]
[s7]
[s8]
[s9]
where [s7] is actually a limit as
,
which we evaluated by applying
rule twice.
_small.jpg)
,
which we evaluated by applying
rule twice.