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

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

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