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]

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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

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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 ]

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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!]

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