Value-at-Risk Theory and Practice

The first advanced book on value-at-risk

Chapter 3, Page 145
Exercise 36

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.

 

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