SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

15
(d)
E[X] = ∑ all a
ap X (a) =0(1− γ)+1(γ)
= γ
(e)
E[X] = ∑ ∞∑
ap X (a) = aγ(1 − γ) a−1
all a a=1
=
∞∑
γ a(1 − γ) a−1 (let q =1− γ)
a=1
= γ
∞∑
a=1
d
dq qa
= γ d ∞∑
q a = γ d dq
a=1
dq
= γ d dq
( ) 1
=
1 − q
∞∑
a=0
q a
(since d dq q0 =0)
γ
(1 − q) = γ
2 (γ) = 1 2 γ
20. From Exercise 25, Var[X] =E[X 2 ]−(E[X]) 2 .SoifwecalculateE[X 2 ], we can combine
it with the answers in Exercise 19.
(a)
E[X 2 ] =
∫ ( β a
2
α
β − α
= β3 − α 3
3(β − α)
)
da =
a 3
3(β − α) ∣
β
α
Therefore,
Var[X] = β3 − α 3 ( ) 2
β + α
3(β − α) − 2
= β3 − α 3 (β + α)2
−
3(β − α) 4