SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

41
(b)
20. (a)
Pr{Y 52 − Y 48 =20,Y 48 =80| Y 52 = 100}
= Pr{Y 52 − Y 48 =20,Y 48 =80,Y 52 = 100}
Pr{Y 52 = 100}
= Pr{Y 52 − Y 48 =20} Pr{Y 48 =80}
Pr{Y 52 = 100}
= Pr{Y 4 =20} Pr{Y 48 =80}
Pr{Y 52 = 100}
= e−2(4) (2(4)) 20
20!
= 100! ( 4
20!80! 52
≈
5 × 10 −5
(c) Pr{Y 2 =0} = e −2(2) ≈ 0.02
(b) 8 a.m. → hour 2
2p.m.→ hour 8
Λ(t) =
Λ(t) =
∫ t
0
∫ t
e −2(48) (2(48)) 80
80!
) 20 ( ) 48 80
=
52
λ(a)da
Λ(t) = Λ(6)+
0
100!
e−2(52) (2(52))100
( )( 100 1
20 13
1da = t, for 0 ≤ t