SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
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16 CHAPTER 3. BASICS<br />
= 4(β3 − α 3 )<br />
12(β − α) − 3(β + α)2 (β − α)<br />
12(β − α)<br />
= 4(β − α)(β2 + βα + α 2 ) − 3(β 2 +2βα + α 2 )(β − α)<br />
12(β − α)<br />
= β2 − 2βα + α 2<br />
12<br />
=<br />
(β − α)2<br />
12<br />
(b)<br />
E[X 2 ] =<br />
=<br />
=<br />
∫ ∞<br />
0<br />
∫ ∞<br />
0<br />
∫ ∞<br />
0<br />
a 2 αβ −α a α−1 e −(a/β)α da<br />
a 2 (α/β)(a/β) α−1 e −(a/β)α da<br />
Using the same substitution as 19(b) gives<br />
∫ ∞<br />
β 2 u 2/α e −u du = β 2 u 2/α e −u du<br />
0<br />
= β 2 Γ(2/α +1)= 2β2<br />
α Γ(2/α)<br />
There<strong>for</strong>e,<br />
Var[X] = 2β2<br />
α<br />
Γ(2/α) − (β/α Γ(1/α))2<br />
= β2<br />
α (2Γ(2/α) − 1/α (Γ(1/α))2 )<br />
(c)<br />
E[X 2 ] = 1 2 (0.3) + 2 2 (0.1) + 3 2 (0.3) + 4 2 (0.25) + 6 2 (0.05)<br />
= 9.2<br />
There<strong>for</strong>e, Var[X] =9.2 − (2.7) 2 =1.91<br />
(d) E[X 2 ]=0 2 (1 − γ)+1 2 (γ) =γ<br />
There<strong>for</strong>e, Var[X] =γ − γ 2 = γ(1 − γ)<br />
(e)<br />
E[X 2 ] =<br />
∞∑<br />
a 2 γ(1 − γ) a−1<br />
a=1<br />
=<br />
∞∑<br />
γ a 2 q a−1 (let q =1− γ)<br />
a=1