STOCHASTIC PROCESS [Kazemi ]- Assignment 1
Basic concepts and definitions
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Exercise 1) Let (Ω, F, P ) be a probability space. Let ρ be a function
defined by ρ(A, B) = P (A 4 B), where A 4 B = (A − B) ∪ (B − A). Show
that, for any C ∈ F,
ρ(A, B) ≤ ρ(A, C) + ρ(B, C).
Exercise 2) Show that for any two random events A and B,
|P (A) − P (B)| ≤ ρ(A, B).
Exercise 3) Let X1 and X2 be independent real-valued r.v.’s on a
probability space (Ω, F, P ) with the distribution functions H1 (.) and
H2 (.), respectively. Let (ξ(t); t ≥ 0) be a stochastic process defined by
ξ(t) = tX1 + X2 , Calculate P (A), where A is the set of all nondecreasing
sample functions of the process.
Exercise 4) Let (ξ(t); t ≥ 0) be a stochastic process defined by ξ(t) =
X + αt (α > 1). Let D ⊂ [0, ∞) be finite or countably infinite. Deremine
a. P {ξ(t) = 0 f or at least one t ∈ D},
b. P {ξ(t) = 0 f or at least one t ∈ (1, 2]}.
Exercise 5) Let X and Y be independent r.v.’s on a probability space
(Ω, F, P ), where Y ∼ N (0, 1). Let (ξ(t); t ≥ 0) be a stochastic process
defined by
ξ(t) = X + t(Y + t).
Let A be the set of sample functions of ξ(t) nondecreasing on [0, ∞).
Calculate P (A).
Exercise 6) Let X1 and X2 be independent r.v.’s on a probability
space (Ω, F, P ) with common standard normal d.f. Let (ξ(t); t ≥ 0) be a
stochastic process defined by
ξ(t) = (X1 + X2 )t.
Determine Ft1 ,...,tn (x1 , ..., xn ). If A is the set of all non-negative sample
functions, determine P (A).
Exercise 7) Let (ξ(t); t ≥ 0) be a stochastic process defined by
ξ(t) = X cos(t + U ),
where X and U are independent r.v’s. U is uniform in [−π, π] and
E[X] = 0. Determine its covariance function.
Exercise 8) Let (ξ(t); t ≥ 0) and (ζ(t); t ≥ 0) be a real stochastic
processes on (Ω, F, P ). If they are stochastically equivalent, show that
they have identical marginal distributions.
Exercise 9) Let {Xi }∞
and {Yi }∞
be r.v’s on (Ω, F, P ) such that
1
1
1
2
E[Xi ] = E[Yi ] = 0, Var[Xi ] = Var[Yi ] = σi2 < ∞, E[Xi Xj ] = 0, E[Yi Yj ] =
0 and E[Xi Yj ] = 0 for all i 6= j . Let (ξ(t); t ≥ 0) be defined by
n n
o
X
ξ(t) =
Xj cos λj t + Yj sin λj t .
j=1
Is the process wide sense stationary(weakly or second-order stationary)? Determine its covariance function.
Exercise 10) Let (ξ(t); t ≥ 0) be a process with independent increments.
Assume that
ϕ1 (t, λ) = E[eiλξ(t) ] , ϕ2 (t1 , t2 , λ) = E[eiλ[ξ(t2 )−ξ(t1 )] ]
are given. Determine the characteristic function of (ξ(t1 ), ..., ξ(tn )) in terms
of ϕ1 and ϕ2 .
Exercise 11) Let (ξ(t); t ∈ T ) be stochastically continuous at every t ∈ T .
Then Ψ(ξ(t)) is also stochastically continuous if Ψ : R → R is continuous.
Exercise 12) In the previous problem, show that ϕ(t) = E[Ψ(ξ(t))] is a
continuous function.
Exercise 13) Let (Xn ; n ≥ 0) be a sequence of i.i.d r.v with mean 0 and
variance 1. Is the {Xn } wide sense stationary?
Exercise 14) Let (ξ(t); t ≥ 0) be a stochastic process defined by
ξ(t) = t + X,
where X ∼ N (0, 1). Determine
a. P {ξ(t) = 0 f or at least one t ∈ D = {t1 , t2 , ...} ⊂ T = [0, ∞]}.
b. P {ξ(t) = 0 f or at least one t ∈ (0, 1]}.
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