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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19<br />
=<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
a 2 f X (a)da − 2E[X] af X (a)da<br />
−∞<br />
∫ ∞<br />
+(E[X]) 2 f X (a)da<br />
−∞<br />
= E[X 2 ] − 2E[X]E[X]+(E[X]) 2<br />
= E[X 2 ] − (E[X]) 2<br />
The proof <strong>for</strong> discrete-valued X is analogous.<br />
26.<br />
E[X] =<br />
=<br />
=<br />
=<br />
∞∑<br />
a Pr{X = a}<br />
a=0<br />
{<br />
∞∑ a<br />
}<br />
∑<br />
Pr{X = a}<br />
a=1 i=1<br />
⎧<br />
⎫<br />
∞∑ ∞∑<br />
∞∑ ⎨ ∞∑<br />
⎬<br />
Pr{X = a} = Pr{X = a}<br />
⎩<br />
⎭<br />
i=1 a=i<br />
i=0 a=i+1<br />
∞∑<br />
Pr{X >i}<br />
i=0<br />
27. E[X m ]=0 m (1 − γ)+1 m γ = γ<br />
28. Notice that<br />
E[I(X i ≤ a)] = 0 Pr{X i >a} +1Pr{X i ≤ a}<br />
= F X (a)<br />
There<strong>for</strong>e,<br />
E[ ̂F X (a)] = 1 m∑<br />
E[I(X i ≤ a)]<br />
m<br />
i=1<br />
= 1 m mF X(a) =F X (a)<br />
29.<br />
E[g(Y )] =<br />
=<br />
∫ ∞<br />
−∞<br />
∫<br />
A<br />
g(a)f Y (a)da<br />
1 f Y (a)da +0<br />
= Pr{Y ∈A}