TAMS32/TEN1 STOKASTISKA PROCESSER
TENTAMEN TORSDAG 25 AUGUSTI 2022 KL 14.00-18.00.
Examinator och jourhavande lärare: Torkel Erhardsson, tel. 28 14 78.
Permitted exam aids: Formel–och tabellsamling i TAMS32 Stokastiska processer (handed
out during the exam). Mathematics Handbook for Science and Engineering (formerly
BETA), by L. Råde och B. Westergren. Calculator with empty memories.
The exam consists of 6 problems worth 3 points each. Grading limits : 8 points for grade 3,
11.5 points for grade 4, 15 points for grade 5. The results will be communicated by email.
Problem 1
A stationary Gaussian process {X(t); t ∈ R}, with mean µX = 0 and spectral
density
SX (f ) = 1
∀f ∈ R,
is the input signal to a stable LTI with impulse response
(
2e−2t , for t ≥ 0;
h(t) =
0,
otherwise.
Let {Y (t); t ∈ R} denote the output signal.
(a) Compute the autocorrelation function of {Y (t); t ∈ R}.
(b) Compute P (Y (1) > Y (0) + 1).
Problem 2
Let {Xt ; t = 0, 1, . . .} be a Markov chain with state space SX = {1, 2, 3},
initial distribution p, and transition matrix P , where


 
0.25 0.75 0
0.2
P =  0.4 0.4 0.2 .
p = 0.6 ,
0.9
0 0.1
0.2
(a) Does the chain have a stationary and asymptotic distribution? The answer must be supported by an argument. If the answer is yes, compute this
distribution.
(b) Compute P (X3 = 1, X2 = 3|X1 = 2, X0 = 2).
(c) Compute P (X3 = 1, X2 = 3|X1 6= 2, X0 = 2).
Problem 3
Let X and Y be random variables such that X has the Exponential(1) distribution, while Y has the conditional pdf
(
xe−xy , for y > 0;
fY |X=x (y) =
0,
otherwise.
Compute the MMSE (the best predictor in the mean square sense) of X
based on Y .
Problem 4
Let {Yt ; t ∈ Z} be the wide sense stationary solution to the AR(1) equation
Yt − aYt−1 = Xt
∀t ∈ Z,
where {Xt ; t ∈ Z} is i.i.d. white noise with mean 0 and variance σ 2 < ∞,
and 0 < |a| < 1. Since the noise is i.i.d., Xt is independent of (Y0 , . . . , Yt−1 )
for each t ∈ Z (you don’t have to show this). Define the random sequence
{Ut ; t = 0, 1, . . .} by:
Yt
Ut = t
∀t = 0, 1, . . .
a
Is {Ut ; t = 0, 1, . . .} a martingale? The answer must be supported by an
argument.
Problem 5
A particle moves along the real line. It starts at a random position X1 , and has
a random velocity X2 . The random variable (X1 , X2 ) has a two-dimensional
Gaussian distribution, with the following mean vector and covariance matrix:
64 −8
10
.
,
CX =
µX =
−8 4
3
Let Y (t) = X1 + tX2 be the position of the particle at time t ≥ 0.
(a) Compute the expectation function and the autocovariance function for
the process {Y (t); t ≥ 0}. Is the process wide sense stationary?
(b) Find a time t1 ≥ 0 such that Y (t1 ), the position of the particle at time
t1 , is independent of the starting position Y (0).
Problem 6
Let {X(t); t ≥ 0} be a Wiener process (Brownian motion) with variance
parameter 0 < σ 2 < ∞. Let
Z n
X(t)
dt
∀n = 1, 2, . . .
Yn =
t2
1
Does {Yn ; n = 1, 2, . . .} converge in mean square as n → ∞? Use a suitable
criterion to prove or disprove.