Course: Stochastic Processes II - M375T Instructor: Gerard Brunick
Problem Set 1.01
1. Let N(t) =
1
[1,∞)(t) +
1
[2,∞)(t) +
1
[3,∞)(t) and compute f(t) = Rt
0s2dN(s).
2. Let Nbe defines as above and let M(t) = N(t)−t. Compute g(t) = Rt
0N(s) dM(s).
3. Let Mbe defined as above, and let Θ(t) =
1
(1,∞)(t) +
1
(2,∞)(t) +
1
(3,∞)(t). Compute
h(t) = Rt
0Θ(s) dM(s).
4. Let Ω = {ω1, ω2, ω3, ω4}, and let C={ω1, ω2, ω3},{ω2}. Compute σ(C) explicitly.
5. Let Ω = Rand let C=(−∞,5),[0,10]. Compute σ(C) explicitly.
6. In this problem we let Ω = R2, so a point in Ω is really a pair of numbers which we
will write as (ω1, ω2)∈Ω. Let X(ω1, ω2) =
1
[0,∞)(ω1), and let Y(ω1, ω2) =
1
[0,∞)(ω2).
We let σ(X, Y ) denote the smallest σ-field that contains all sets of the form {X≤x}
and {Y≤y}for constants xand y. Compute σ(X), σ(Y), and σ(X , Y ) explicitly.
7. Let Fand Gbe σ-fields over some set Ω. Show that F∩Gis a σ-field.
8. What is the smallest set that contains an even number and the number 7? Does this
question even make sense? We defined σ(C) to be the smallest σ-field which contains
all the sets in C. Is this object well-defined?
9. Let (Ω,F,P) be a probability space and let Xbe a random variable, so {X≤x} ∈ F
for any constant x. Show that {x≤X≤y} ∈ Fand {x < X < y} ∈ Ffor any
constants xand y.
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