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Probability and Stochastic Processes

35361 Probability and Stochastic Processes, Spring 2015
Assignment 1 (due to September 9, 2015)
The assignment should be handed in to Tim Ling on 9/09/2015. You
may use any software for calculations (Mathematica is preferable).
Problem 1.
1) Let Xi
, i = 1, 2 be independent random variables (i.r.v.’s) having a
Chi-square distribution, EXi = 4.
(i) Using characteristic functions and the inversion formula find the
probability density function (pdf) of Y = X1 − X2.
(ii) Find E(Y
8).
1
Problem 2. 1) Using a variance reduction technique (e.g. control
variates) find the Monte-Carlo approximation for the integral
J =
Z ∞
0
e
−x
2
(1 + x
2
)dx.
Use the sample sizes n = 106
, n = 107 and compare the results with
the exact value.
2) Using the 3-sigma rule estimate a sample size n required for obtaining
a Monte-Carlo approximation with a control variate for J with
an absolute error less than ∆ = 10−6.
2
Problem 3.
Let B0(t), t ∈ [0, 1] be a Brownian Bridge that is a Gaussian process
with E(B0(t)) = 0 and the covariance function
R(t, s) = min(t, s) − ts.
1) Using simulations with a discrete-time process approximation for
B0(t) (e.g. use N=1000 trajectories and n=1000 discretisation points)
find an approximation for the distribution function of the random variable
X = max
0≤t≤1
|B0(t)|
at the points {0.2, 0.6, 2.0}. Hint: use a representation for B0(t) in
terms of a standard Brownian motion.
2) Verify the results using the analytical expression for the distribution
function of X :
P{X < x} = 1 + 2 X ∞ k=1 (−1)k e −2k 2x 2

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Probability and stochastic processes;

Probability and stochastic processes;

The Questions
1.
Random variables.
a) We place uniformly at random n = 200 points in the unit interval [0, 1]. Denote
by random variable X the distance between 0 and the first random point on the
left.
i)
Find the probability distribution function FX(x).
[3]
ii) Derive the limit as ?? ? 8 and comment on your expression.
[3]
b) The random variable X is uniform in the interval (0, 1). Find the density function of
the random variable Y = – lnX.
[4]
c) X and Y are independent, identically distributed (i.i.d.) random variables with
common probability density function
???? (??) = ?? -?? ,
???? (??) = ??
-??
,
??>0
??>0
Find the probability density function of the following random variables:
i)
Z = XY.
[5]
ii)
Z = X / Y.
[5]
iii)
Z = max(X, Y).
[5]
Probability and Stochastic Processes
© Imperial College London
page 2 of 5
2.
Estimation.
a)
The random variable X has the truncated exponential density
??(??) = ???? -??(??-??0 ) , ?? > ??0 . Let x0 = 2. We observe the i.i.d. samples xi = 3.1,
2.7, 3.3, 2.7, 3.2. Find the maximum-likelihood estimate of parameter c.
[8]
b)
Consider the Rayleigh fading channel in wireless communications, where the
channel coefficients Y(n) has autocorrelation function
???? (??) = ??0 (2?????? ??)
where J0 denotes the zeroth-order Bessel function of the first kind (the function
besselj(0,.) in MATLAB), and fd represents the normalized Doppler frequency
shift. Suppose we wish to predict Y(n+1) from Y(n), Y(n – 1), …, Y(1). The
coefficients of the linear MMSE estimator
??(?? + 1) = ?
??
???? ??(??)
??=1
are given by the Wiener-Hopf equation
???? = ??
where ?? = [??1 , ??2 , … , ???? ]?? , ?? = [???? (??), ???? (?? – 1), … , ???? (1)]?? , and R is a n-byn matrix whose (i, j)th entry is ???? (?? – ??).
i)
Give an expression for the coefficient of the first-order MMSE estimator,
i.e., n = 1.
[4]
ii)
Let fd = 0.01. Write a MATLAB program to compute the coefficients of
the n-th order linear MMSE estimator and plot the mean-square error
????2 = ??0 – ??* ??-?? ?? as a function of n, for 1 = ?? = 20.
[10]
iii)
From the figure, determine whether Y(n) is a regular stochastic process or
not, and justify.
[3]
[As you may imagine, n cannot be greater than 2 for computation of this kind in an
exam.]
Probability and Stochastic Processes
© Imperial College London
page 3 of 5
3.
Random processes.
a) The number of failures N(t), which occur in a computer network over the time interval [0, t),
can be modelled by a Poisson process {N(t), t = 0}. On the average, there is a failure after
every 4 hours, i.e. the intensity of the process is equal to ? = 0.25.
i)
What is the probability of at most 1 failure in [0, 8), at least 2 failures in [8, 16), and at
most 1 failure in [16, 24) ? (time unit: hour)
[7]
ii) What is the probability that the third failure occurs after 8 hours?
[4]
b) Find the power spectral density S( ) if the autocorrelation function
i)
2
??(??) = ?? -???? .
2
ii) ??(??) = ?? -???? cos(
c)
[3]
0 ??) .
[3]
The random process X(t) is Gaussian and wide-sense stationary with E[X(t)] = 0. Show that if
2 (??).
??(??) = ?? 2 (??), then autocovariance function ?????? (??) = 2??????
[8]
Hint: For zero-mean Gaussian random variables Xk,
??[??1 ??2 ??3 ??4 ] = ??[??1 ??2 ]??[??3 ??4 ] + ??[??1 ??3 ]??[??2 ??4 ] + ??[??1 ??4 ]??[??2 ??3 ]
Probability and Stochastic Processes
© Imperial College London
page 4 of 5
4.
Markov chains and martingales.
a)
Classify the states of the Markov chain with the following transition matrix
0
1/2 1/2
0
1/2)
?? = (1/2
1/2 1/2
0
[2]
??=
(
1/2 1/2
0
0
0
1/2 1/2
0
2/3
0
0
1/3
0
0
2/3 1/3
1/3 1/3
0
0
0
0
0
0
1/3)
[3]
b)
Consider the gambler’s ruin with state space E = {0,1,2,…,N} and transition
matrix
1
0
?? 0 ??
?? 0 ??
??=
.
. .
?? 0 ??
0
(
1)
where 0 < p < 1, q = 1 – p. This Markov chain models a gamble where the
gambler wins with probability p and loses with probability q at each step.
Reaching state 0 corresponds to the gambler’s ruin.
i)
?? ????
Denote by Sn the gambler’s capital at step n. Show that ???? = (??)
is a
martingale (DeMoivre’s martingale).
[4]
ii) Using the theory of stopping time, derive the ruin probability for initial
capital i (0 < i < N).
c)
[4]
Let N = 10. Write a computer program to simulate the Markov chain in b).
Starting from state i and run the Markov chain until reaching state 0. Repeat it for
100 times, and plot the ruin probabilities as a function of the gambler’s initial
capital i (0 < i < N), for
i)
p = 1/3;
[4]
ii) p = 1/2;
[4]
iii) p = 2/3.
[4]
Also plot the theoretic results of b).
[Obviously, such a question cannot be tested in this way in the exam!]
Probability and Stochastic Processes
© Imperial College London
page 5 of 5

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  simple order process Pricing order download You have just landed to the most confidential, trustful essay writing service to order the paper from. We have a dedicated team of professional writers who will assist you with any kind of assignment that you have. Your work is handled confidentially with a specific writer who ensures you are satisfied. Be assured of quality, unique and price friendly task delivered in good time when you place an order at our website. banner We have a team of dedicated writers with degrees from all spheres of study. We readily accept all types of essays and we assure you of content that will meet your expectations. Our staff can be reached  via live chat, email or by phone at any given time with prompt response.  

Prices starting at:

High school
Undergrad.(yrs 1-2)
Undergrad.(yrs 3-4)
Master’s
Doctoral
$10 / page
$10/ page
$12/ page
$15 / page
$17/ page

Note: The prices in the table above are applicable to orders completed within 14 days. Kindly see the full price table for more prices..

We urge you to provide as much information as possible to avoid many revisions. Set your deadline, choose your level, give payment information and relax while you track your work. We will deliver your paper on time. . 1 Are you in a quagmire and you are unable to complete your assignments? Do you doubt yourself on the quality of the essay you have written? Myprivateresearcher.com  is here to assist you. We will offer you unmatched quality that is plagiarism free. Place an order at myprivateresearcher.com  for guaranteed  high grades. order download

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