EXAM
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MA430-G
Probability Theory and Stochastic Processes
December 13, 2016
4 hours
6
Pocket calculator,
Lecture notes (as in Fronter).
Notes:
1
Problem 1 (10 points)
Let A and B be two events with probabilities P (A) = a and P (B) = b,
respectively, where a and b are real-valued constants 0 < a, b ≤ 1. Determine
the conditional probability P (A|B) if
1.1) A and B are mutually exclusive.
1.2) A and B are independent.
1.3) A ⊂ B.
1.4) B ⊂ A.
1.5) P (A ∩ B) = c, where c is a real-valued constant and 0 < c ≤ a .
Problem 2 (10 points)
An experiment is defined by flipping an unfair coin three times, where the result
of each flip is independent of any preceding result. The probability of coming
up heads h and tails t equals P (h) = 2/3 and P (t) = 1/3, respectively.
2.1) Determine all possible outcomes and their corresponding probabilities.
2.2) Find the probability P (A) of the event A that the first flip results in a head.
2.3) Find the probability P (B) of the event B that the number of heads is even.
2.4) Find the probability P (A ∩ B) of the events A and B.
2.5) Find the conditional probability P (A|B) of the events A and B.
2
Problem 3 (15 points)
The probability density function (PDF) fX (k) of a discrete random variable X
is given by
fX (k) = P {X = k} =



a(k 2 + 4) ,
k = 0, 1, 2, 3, 4,


0,
otherwise,
(3.1)
where a is a real-valued constant.
3.1) Determine the constant a and sketch the PDF fX ( k) of X.
3.2) Find and sketch the corresponding cumulative distribution function (CDF)
FX (k) of X.
3.3) Find the probability P {X > 1}.
3.4) Find the conditional probability P {X = 3|X ≥ 2}.
3.5) Find the mean E{X} of X.
3.6) Find the variance Var{X} of X.
Problem 4 (10 points)
Let X be a uniformly distributed random variable over the interval [1, 3].
6
.
Given is another random variable Y , which is defined as Y = X+2
4.1) Sketch the PDF fX (x) of X.
4.2) Find the PDF fY (y) of Y .
4.3) Sketch the PDF fY (y) of Y .
3
Problem 5 (20 points)
Let X and Y be two random variables, which are characterized by the joint PDF
fXY (x, y) =



xy , if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2,


0,
otherwise.
(5.1)
5.1) Find the marginal PDF fX (x) of X.
5.2) Find the marginal PDF fY (y) of Y .
5.3) Are X and Y dependent or independent? Explain your answer.
5.4) Are X and Y correlated or uncorrelated? Explain your answer.
5.5) Find the covariance CXY of the two random variables X and Y .
5.6) Find the probability P {X 2 + Y 2 ≤ 1}.
Hints:
1. The Cartesian coordinates (x, y) can be transformed into polar coordinates
(r, θ) by means of x = r cos(θ), y = r sin(θ), and dx dy = r dr dθ.
2. sin(α) cos(β) = 21 [sin(α + β) + sin(α − β)]
3.
Z
1
sin(ax)dx = − cos(ax)
a
4
Problem 6 (15 points)
Let Xi (i = 1, 2, . . . , n) be the lifetime of a light bulb, which is used until it
fails, and then it is replaced by a new light bulb. The lifetimes Xi of the light
bulbs are independent and identically distributed (i.i.d.) random variables with
2
mean E{Xi } = µX = 100 h (in hours) and variance Var{Xi } = σX
= 25 h2 .
Another random variable Y is defined as the total lifetime of n light bulbs,
which are used one-by-one, i.e., Y = X1 + X2 + · · · + Xn .
6.1) Find the mean E{Y } and the variance Var{Y } of Y .
6.2) Give reasons, why the cumulative distribution function (CDF) FY (y) of Y
can be expressed by


y − nµX
FY (y) = G  q 2  if n → ∞
nσX
where G(·) is the CDF of the standard normal random variable.
6.3) For n = 36, find an approximate expression for the probability that Y is
between 3500 and 3700, i.e., P {3500 ≤ Y ≤ 3700}.
Hint: Some useful values of the CDF G(x) of the standard normal random
variable:
G(1) = 0.6
G(3) = 0.7
G(10/3) = 0.9 .
5
Problem 7 (20 points)
Let X(t) = e At be a stochastic process, where A is a real-valued random
variable, which is uniformly distributed between 1 and 3.
7.1) Compute the mean µX (t) of X(t) by using µX (t) = E{X(t)}.
7.2) Compute the autocorrelation function RXX (t1 , t2 ) of X(t) by using
RXX (t1 , t2 ) = E{X(t1 )X ∗ (t2 )}.
7.3) Is the stochastic process X(t) wide-sense stationary (WSS)? Give reasons
for your answer.
7.4) Compute the time average µX of a single sample function X(t; ai ) using
µX =< X(t; ai ) >, where ai is a constant (outcome of A) and < · >
denotes the time averaging operator.
7.5) Is the stochastic process X(t) mean-ergodic? Give reasons for your answer.
7.6) Compute the autocorrelation function RXX (τ ) of a single sample function
X(t; ai ) using RXX (τ ) =< X(t + τ ; ai )X ∗ (t; ai ) >.
7.7) Is the stochastic process X(t) autocorrelation ergodic? Give reasons for
your answer.
Hint:
Z
eax
,
e dx =
a
ax
where a 6= 0 is a constant.
6
(7.1)