Download TAMS32/TEN1 STOKASTISKA PROCESSER TENTAMEN L

TAMS32/TEN1 STOKASTISKA PROCESSER
TENTAMEN LÖRDAG 7 JANUARI 2017 KL 08.00-12.00.
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.
Uppgift 1
Let {Xt ; t = 0, 1, . . .} be a Markov chain with state space SX = {0, 1, 2},
initial distribution p(0) and transition matrix P , where


 
1/3 a 2/3
1/2
b .
P = 1/2 0
p(0) = 1/2 ,
c 3/4 0
0
(a) Compute the constants a, b and c.
(b) Compute p(2), that is, compute the probabilities P (X2 = i), i = 0, 1, 2.
(c) Does the chain have a stationary distribution? Does it have an asymptotic
distribution? Motivate your answers. Compute these distributions if they
exist.
Uppgift 2
Let {B(t); t ≥ 0} be a Wiener process (Brownian motion) with variance
parameter σ 2 = 1. The Brownian bridge process {X(t); 0 ≤ t ≤ 1} is defined
by:
X(t) = B(t) − tB(1)
∀0 ≤ t ≤ 1.
(a) Compute P (−1 < X( 21 ) < 1).
(b) Let Y = X( 21 ) − X(0) and Z = X(1) − X( 12 ). Compute the correlation
coefficient ρ(Y, Z).
(c) Does the Brownian bridge have independent increments? Motivate your
answer.
Uppgift 3
Let {X(t); t ∈ R} be a white noise process in continuous time, with spectral
density
SX (f ) = 1
∀f ∈ R.
Decide whether a wide sense stationary process {Y (t); t ∈ R} with mean
µY = 0, and autocovariance function CY (τ ) = e−2|τ | ∀τ ∈ R, can be obtained
as the output signal of a linear time invariant filter (LTI), when the input
signal is the white noise {X(t); t ∈ R}. Motivate your answer. If the answer
is yes, give the impulse response for such an LTI.
Uppgift 4
Let {Yt ; t ∈ Z} be the wide sense stationary AR(2) process which solves the
equation
1
1
Yt − Yt−1 − Yt−2 = Xt
∀t ∈ Z,
6
6
where {Xt ; t ∈ Z} is white noise with E(Xt ) = 0 and V (Xt ) = 1. The
autocorrelation function of {Yt ; t ∈ Z} is
RY (τ ) =
48 1 |τ | 27 1 |τ |
( ) + (− )
70 2
70 3
∀τ ∈ Z.
(You do not have to show this.)
(a) Compute the LMMSE of Yt based on (Yt−1 , Yt−2 ), and the corresponding
mean square prediction error.
(b) Compute the LMMSE of Yt based on Yt−1 , and the corresponding mean
square prediction error.
Uppgift 5
Let X and Y be√independent
random variables, such that X ∼ N(0, 1), and
√
Y ∼
3, 3), that is, Y is uniformly distributed over the interval
√ Uniform(−
√
(− 3, 3). Let
Z(t) = X cos(ωt) + Y sin(ωt)
∀t ∈ R,
where ω > 0 is a constant. Is the stochastic process {Z(t); t ∈ R} wide sense
stationary? Is it stationary? Motivate your answer(s).
Hint: It might be useful to know that cos(α+β) = cos(α) cos(β)−sin(α) sin(β),
and that cos(α − β) = cos(α) cos(β) + sin(α) sin(β).
Uppgift 6
A bus starts from its terminus at time T0 = 0, with 10 passengers. The
bus lets off one passenger at a time, at times 0 < T1 < T2 < · · · < T10 ,
which are given by a Poisson process {N (t); t ≥ 0} with intensity 1. (That is:
0 < T1 < T2 < · · · < T10 are the first 10 jump times of the Poisson process.)
Let Xi (i = 1, . . . , 10) be the time it takes for passenger no. i to reach his/her
home after he/she has disembarked from the bus. Assume that the random
variables X1 , X2 , . . . , X10 are independent and Exponential(λ) distributed,
and also independent of the Poisson process. Compute the probability that
none of the 9 first passengers have reached their homes at time T10 (when
the last passenger is let off from the bus). Compute also the numerical value
of this probability when λ = 0.1.
Hint: Compute first the conditional probability that none of the passengers have reached their homes at time T10 , given the event {T1 = t1 , T2 =
t2 , . . . , T10 = t10 }. Then, use some suitable properties of the random variables T1 , T2 , . . . , T10 .