. (a) Let X follow an exponential distribution with parameter = 1. Denote the mean value and variance of X by u =1 and o? = 1 respectively. i. Compute the approximate expectation of X3 using the 2nd order moment approximation E#(X)] ~ o(µ) + }ø"(µ)o² with (x) = x³. Sketch the graph of o as well as the approximating function obtained from 2nd order Taylor approximation about u, i.e. about x = 1. ii. Compute E[X*] exactly, e.g. by integration or by using the mgf. [TYPE:] Discuss the direction of the deviation of the approximation computed above from the true value. (b) Let W and Z be independent both following a U(0, 7/2) distribution. i. Compute E[sin(W+Z)] using the 2nd order moment approximation. Hint:
. (a) Let X follow an exponential distribution with parameter = 1. Denote the mean value and variance of X by u =1 and o? = 1 respectively. i. Compute the approximate expectation of X3 using the 2nd order moment approximation E#(X)] ~ o(µ) + }ø"(µ)o² with (x) = x³. Sketch the graph of o as well as the approximating function obtained from 2nd order Taylor approximation about u, i.e. about x = 1. ii. Compute E[X*] exactly, e.g. by integration or by using the mgf. [TYPE:] Discuss the direction of the deviation of the approximation computed above from the true value. (b) Let W and Z be independent both following a U(0, 7/2) distribution. i. Compute E[sin(W+Z)] using the 2nd order moment approximation. Hint:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. (a) Let X follow an exponential distribution with parameter A = 1. Denote the mean
value and variance of X by pu = 1 and o? =1 respectively.
i. Compute the approximate expectation of X using the 2nd order moment
approximation E[¢(X)]
graph of o as well as the approximating function obtained from 2nd order
Taylor approximation about u, i.e. about r = 1.
ii. Compute E[X³] exactly, e.g. by integration or by using the mgf. [TYPE:]
Discuss the direction of the deviation of the approximation computed above
from the true value.
2 ø(u) + }0"(H)² with (x) = x³. Sketch the
(b) Let W and Z be independent both following a U(0, T/2) distribution.
i. Compute E[sin(W +Z)] using the 2nd order moment approximation. Hint:
write V = W +Z. What are E[V] and Var(V)?
ii. For functions (:) and (-), write down the definition of Elo(W)(Z)] as
an integral. By considering the form of this integral, explain carefully why
E[¢(W)¼(Z)] = E[ø(W)]E{v(Z)].
iii. Compute E[sin(W + Z)] exactly. How accurate is the 2nd-order moment
approximation from part (b)i compared to the one in part (a)? Hint: Re-
membering trigonometric identities will help you use the result from part (b)ii
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Transcribed Image Text:1. (a) Let X follow an exponential distribution with parameter A = 1. Denote the mean
value and variance of X by pu = 1 and o? =1 respectively.
i. Compute the approximate expectation of X using the 2nd order moment
approximation E[¢(X)]
graph of o as well as the approximating function obtained from 2nd order
Taylor approximation about u, i.e. about r = 1.
ii. Compute E[X³] exactly, e.g. by integration or by using the mgf. [TYPE:]
Discuss the direction of the deviation of the approximation computed above
from the true value.
2 ø(u) + }0"(H)² with (x) = x³. Sketch the
(b) Let W and Z be independent both following a U(0, T/2) distribution.
i. Compute E[sin(W +Z)] using the 2nd order moment approximation. Hint:
write V = W +Z. What are E[V] and Var(V)?
ii. For functions (:) and (-), write down the definition of Elo(W)(Z)] as
an integral. By considering the form of this integral, explain carefully why
E[¢(W)¼(Z)] = E[ø(W)]E{v(Z)].
iii. Compute E[sin(W + Z)] exactly. How accurate is the 2nd-order moment
approximation from part (b)i compared to the one in part (a)? Hint: Re-
membering trigonometric identities will help you use the result from part (b)ii
here.
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