SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

11
16. Let U be a random variable having the uniform distribution of [0, 1].
Let V =1− U. Then
Pr{V ≤ a} = Pr{1 − U ≤ a}
= Pr{U ≥ 1 − a} = a, 0 ≤ a ≤ 1.
Therefore, V and U are both uniformly distributed on [0, 1].
Therefore, Y = − ln(1 − U)/λ = − ln(V )/λ and Y = − ln(U)/λ must have the same
distribution.
17. (a) U = F (Y )= a−α
β−α
Therefore, Y =(β − α)U + α
algorithm uniform
1. U ← random()
2. Y ← (β − α)U + α
3. return Y
For α =0,β=4
U Y
0.1 0.4
0.5 2.0
0.9 3.6
(b) U = F (Y )=1− e −(Y/β)α
Therefore,
algorithm Weibull
1. U ← random()
2. Y ← β(− ln(1 − U)) 1/α
3. return Y
For α =1/2, β=1
U Y
0.1 0.011
0.5 0.480
0.9 5.302
1 − U = e −(Y/β)α
ln(1 − U) = −(Y/β) α
Y = β(− ln(1 − U)) 1/α