ECE 511 Special Examples Summer 2023
SE 2.1: two urns contain, respectively 2 white and 1 red and 2 white and 2 red balls. A person
randomly transfers one ball from the first urn into the second. A ball is now drawn from the
second urn. What is the probability of it being red?
SE 2.2: Three urns contain, respectively, 2 white and 1 red, 2 white and 2 red, and 1 white and 2
red balls. A person randomly transfers one ball from the first urn into the second, then one ball
from the second into the third. A ball is now drawn from the third urn. What is the probability
of it being red?
SE 2.3: In a die experiment, determine the conditional probability of the event {f2} given that
the event even occurred
SE 2.4: A box contains three white balls w1, w2, and w3 and two red balls r1 and r2. Two balls
are removed at random. Determine the probability that the first removed ball is white and the
second red?
SE 2.5: A box contains 6 white and 4 black balls. Remove two balls at random without
replacement. What is the probability that the first one is white and the second one is black?
SE 2.6: Mary has two children. One child is a boy. What is the probability that the other child is
a girl?
SE 2.7: Two boxes B1 and B2 contain 100 and 200 light bulbs respectively. The first box (B1) has
15 defective bulbs and the second 5. Suppose a box is selected at random and one bulb is
picked out.
(a) What is the probability that it is defective? (a priori)
(b) Suppose we test the bulb and it is found to be defective. What is the probability that it
came from box 1? P (B1 / D) = ?
SE 2.8: We have four boxes. Box 1 contains 2000 components of which 5% are defective. Box 2
contains 500 components of which 40% are defective. Boxes 3 and 4 contain 1000 each with
10% defective. We select at random one of the boxes and we remove at random a single
component
(a) What is the probability that the selected component is defective?
(b) We examine the selected component and we find it defective. On the basis of the evidence,
we want to determine the probability that it came from box 2.
SE 2.9: Suppose there exists a (fictitious) test for cancer with the following properties. Let
A : event that the test states that tested person has cancer
B : event that person has cancer
A* : event that test states person is free from cancer
B* : event that person is free is free from cancer
It is known that P{A / B} = P{A* / B*} = 0.95 and P{B} = 0.005. Is the test a good test?
SE 2.10: Three switches connect in parallel operate Independently. Each switch remains closed
with probability p.
(a) Find the probability of receiving an input signal at the output.
(b) Find the probability that switch S1 is open given that an input
2
(c) Signal is received at the output.
SE 3.1: A box B1 contains 8 white and 6 red balls and box B2 contains 5 white and 15 red balls, A
ball is drawn from each box. What is The probability that the ball from B1 is white and the ball
from B2 is red?
SE 3.2: A fair die is rolled five times. Find the probability p5 (2) that “six” will show twice.
SE 3.3: A pair of dice is rolled n times. (a) Find the probability that “seven” will not show at all
(b) (Pascal) Find the probability of obtaining double six at least once
SE 3.4: There are four balls numbered 1 to 4 in the urn. Determine The number of
distinguishable, ordered samples of size 2 that can be drawn without replacement. Determine
the number of distinguishable unordered sets
SE 3.5: An odd number of people want to play a game that requires two teams made up of
even number of players. To decide who shall be left out to act as umpire, each of N persons
tosses a fair coin with the following stipulations: If there is one person whose outcome (be it
heads or tails) is different from the rest of the group, that person will be the umpire. Assume
that there are 11 players. What is the probability that a player will be “odd-person out”, i.e.,
will be the umpire on the first play?
SE 3.6: 10 independent binary pulses per second arrive at a receiver. The error probability, i.e.,
a zero received as a one or vice versa) is 0.001. What is the probability of at least one
error/second?
SE 3.7: An order of 104 parts is received. The probability that a part is defective equals 0.1.
What is the probability that the total number of defective parts not exceed 1100?
SE 3.8: Two urns contain balls as follows: #1: 3 Red and 4 Black and #2: 2 Red and 1 Black. A
person randomly transfers one ball from the first urn into the second and then one ball from
the second into the first. A ball is now drawn from the first urn. What is the probability of it
being Red?
SE 3.9: Two assembly lines produce balloons. The capacities of the Line #1 and Line #2 are 100
per min. and 50 per min. respectively. The probability of a balloon being defective in line #1 is
2% whereas the corresponding value for line #2 is 5%.
(a) Find the probability that a balloon randomly chosen in the market is defective.
(b) If a randomly chosen balloon in the market is found to be defective, what is the probability
that it came from assembly line #1
SE 4.1: Toss a coin. Ω = {H, T} with P(H) = p. Suppose the r.v X is such that X (T) = 0, X (H) = 1.
Find FX (x).
SE 4.2: A fair coin is tossed twice, and let the r.v X represent the number of heads. P(H) = p. Find
FX (x).
SE 4.3: The pdf of a random Variable x is shown in the figure
3
(a) Compute the value of A
(b) Find Fx (x) and sketch the PDF
(c) Compute P [ 2 ≤ x < 3]
(d) Compute P [ 2 < x ≤ 3]
(e) Compute FX (3)
SE 4.4: The pdf of a random variable is shown in the figure.
(a) Compute the value of A. (b) sketch the PDF; (c) Compute P[2 ≤ X < 3];
(d) Compute P[2 < X ≤ 3]; (e) Compute FX (3).
SE 4.5: Consider the random variable X with pdf fX (x) given by
(a) Find A and plot fX (x)
(b) Plot PDF FX (x)
(c) Find point b such that P [X > b] = ½ P[X ≤ b]
SE 4.6: Let p = P(H) represent the probability of obtaining a head in a toss. For a given coin, a-
priori p can possess any value in the interval (0,1). In the absence of any additional information,
we may assume the a-priori p.d.f fP (p) to be a uniform distribution in that interval. Now
suppose we actually perform an experiment of tossing the coin n times, and k heads are
observed. This is new information. How can we update fP (p) ?
4
SE 4.7: Consider the random variable X with pdf fX (x) as shown in figure below.
(a) Find the value of K
(b) Determine the expression for and plot the PDF FX (x)
(c) Find the probability for X being in the interval -1/2 < X < ½
SE 4.8: A random variable X is defined to be exponential function
fX (x) = (1/ λ) exp (-x/ λ) u(x). The variable Y is a piecewise function of X as shown in the figure
below:
(a) Find and plot the pdf fY (y).
(b) Find and plot the PDF FY (y)
SE 5.1: Let X be a uniform r.v. on (0,1), i,e, X ~ U(0,1) and let Y = 2X + 3. Find fY (y).
SE 5.2: Determine FY (y) and fY (y) if Y = aX + b
SE 5.3: (Square Law Detector) . Solve for FY y) and fY (y) given Y = X2
SE 5.4: If fX (x) is U (-1, 1) and Y = X2, determine FY (y) and fY (y)
SE 5.5: Find fY (y) in terms of fX (x) if





−≤+
≤<−
>−
==
.,
,,0
,,
)(
cXcX
cXc
cXcX
XgY
SE 5.6: The curve g(X) is constant for x ≤ -b and x ≥ b and in the interval (-b, b), it is a straight
line. Sketch FY (y).
SE 5.7: Sketch FY (y) if
5
SE 5.8: The function g(x) equals 0 in the interval ( -c, c) and it is discontinuous for x
= + c with g(c+) = c, g(c-) = 0. Sketch FY (y)
SE 5.9: Half-wave rectifier



≤
>
==
.0,0
,0,
)( );(
x
xx
xgXgY
; = x u(x)
For all yxy −=> 1 ,0 and yx +=2
SE 5.11: Find FY (y) and fy (y) if fX (x) = a e-ax u(x) and Y = X2
SE 5.12: Consider fX (x) to be U(1,2). Determine and sketch fY (y) if Y = (1/X)
SE 5.13: Suppose ,0 ,/2)(
2 ππ <<= xxxf X and .sin XY = Determine ).(yfY
SE 5.14: Let Y = tan X where X ∼ U(- ∏ / 2, ∏ / 2. Determine fY (y).
SE 5.16: Given that x is a normal rv N (0; 1), determine the pdf fY (y) if Y : 2x + 1
for x> 0; x2 + 1 for x ≤ 0
SE 5.17: Let X be a uniform r.v. on [0, 2]. Compute the pdf of Y if Y = g(X) and
g(.) is as shown in the figure
SE 5.18: Suppose X ~ P(lambda) so that
,2,1,0 ,
!
)( === − k
k
ekXP
kλλ
Define Y = X2 + 1 Find the p.m.f of Y.
SE 5.10:(Square Law Detector) Solve for F
Y
y) and f
Y
(y) given Y = X
2
SE 5.15: Find F
Y
(y) and f
y
(y) if f
X
(x) = a e
-ax
u(x) and y = (x + 2 ) u (x + 2)
6
SE 5.19: Given X ~ U(0, 1) and Y = 2X + 3. Find Mean E[Y].
SE 5.20: Let a r.v. X denote the outcome of throwing a fair die. Find the mean and
standard deviation of X
SE 5.21: Find the mean and variance of the exponential density function fX (x) =
(1/λ) e (- x/λ) 0 < x < Ꝏ
SE 5.22: Find the mean and variance of the r.v. X defined as fX (x) = 2x 0otherwise.
SE 5.23: For a Poisson distribution, determine the characteristic
Function and use it to find the mean.
SE 5.24: The characteristic function of a r.v. X is given by ΦX (ω) = 1 - |ω| for |ω|
< 1; 0 for |ω| > 1. Find the pdf of X
SE 5.25: For a random variable X, the probability density function fX (x) is specified
to be uniform U ( -1, 1). A function Y is defined to be y = 3 |x|. Find and sketch the
pdf fY (y) and PDF FY (y).
SE 5.26: Find and sketch the PDF FY (y) and pdf fY (y) if PDF FX (x) = ( 1 – e-2x ) U(x).
and y = x2.
Hint: If you need to calculate ∫ f ( t1/2 ) dt, use change of variable t1/2 = u
SE 5.27: Find and plot fY (y) given that the random variable X is uniformly
distributed in the range -2 < x < 2 and the function y is defined as
-1 x ≤ -1
y = x -1 < x < 1
( 2 – x) x > 1
Hint:
7
SE 5.28: The characteristic function of a random variable is defined to be
5 / (5 – j ω). Find the mean and variance of the function.
SE 6.1: Given


 <<<
=
. otherwise0,
,10K,
),(
yx
yxf XY
, Obtain the marginal p.d.fs )(xf X and ).(yfY
SE 6.2: Given
otherwise.,0
,10 ,0,),(
2


 <<∞<<
=
− xyexy
yxf
y
XY
Determine whether X and Y are independent.
SE 6.6: Z = X + Y. Find ).(zfZ
SE 6.7: Suppose X and Y are independent exponential r.vs with common
parameter λ, and let Z = X + Y. Determine ).(zfZ
SE 6.8: X and Y are independent uniform r.vs in the common interval (0,1).
Determine ),(zfZ where Z = X + Y.
8
SE 6.9: Let .YXZ −= Determine its p.d.f ).(zfZ
SE 6.11: (Square Law Detector). Let X and Y be independent, uniform r.v.s in
(-1, 1). Compute the pdf of V = (X + Y )2.
SE 6.12: Given Z = X / Y, obtain its density function
SE 6.13: X and Y are jointly normal random variables with zero mean so that
.
12
1),(
2
2
2
21
2
1
2
2
2
)1(2
1
2
21








+−
−
−
−
= σσσσ
σπσ
yrxyx
r
XY e
r
yxf
. Find the density function for Z = X/Y.
SE 6.14: .22 YXZ += Obtain ).(zfZ
SE 6.15: X and Y are independent normal r.vs with zero Mean and common
variance .2σ Determine )(zfZ for .22 YXZ +=
SE 6.16: Let .22 YXZ += Find ).(zfZ
SE 6.17: Redo example SE 6.15, where X and Y have nonzero means Xµ and Yµ
respectively.
SE 6.18: (Discrete Case): Let X and Y be independent Poisson random variables
with parameters 1λ and 2λ respectively. Let .YXZ += Determine the p.m.f of Z.
SE 6.19:
FVW (v, w) and fVW (v, w).
9
SE 6.20:
SE 6.21
SE 6.22: Let X and Y be independent exponential random variables with
common parameter λ. Define U = X + Y, V = X - Y. Find the joint and marginal
p.d.f of U and V.
SE 6.23: Suppose Z = X + Y and let W = Y so that the transformation is one-to-one
and the solution is given by . , 11 wzxwy −== . Find fZ (z).
SE 6.24: Let X and Y be independent standard Normal Distributions N (0, 1). Find
the pdf of Z = X/Y. Note that
. ,),(||
),()(),()(
0
0
+∞<<∞−=
−+=
∫
∫∫
∞+
∞−
∞−
∞+
zdyyyzfy
dyyyzfydyyyzyfzf
XY
XYXYZ
SE 6.25: Find the pdf fz(z) if Z = (X2 + Y2)1/2 using the auxiliary Variable method. X
and Y are specified to be independent Gaussian functions N (0, σ). (ref SE 6.16)
SE 6.26:
with probabilities ¼, 1/2., and ¼ respectively. Find if they are uncorrelated and/or Independent
Assume X and Y are N (0, 1) and independent
10
SE 6.27: Let X ∼ U(0,1) , Y ∼ U(0.1). Suppose X and Y are independent. Define Z =
X + Y, W = X - Y . Show that Z and W are dependent, but uncorrelated r.vs.
SE 6.28: Let .bYaXZ += Determine the variance of Z in terms of YX σσ ,
and .XYρ
SE 6.29: Consider two i.i.d. r.v.’s X and Y with the following pdfs. Compute the pdf
of Z = X + Y using characteristic functions.
SE 6.30: Let X and Y be jointly Gaussian r.vs with parameters ).,,,,(
22 ρσσµµ YXYXN
Define .bYaXZ += Determine ).(zfZ
SE 6.31: Suppose X and Y are jointly Gaussian r.vs as in the previous example.
Define two linear combinations . , dYcXWbYaXZ +=+= what can we say about
their joint distribution?
SE 6.32: Two fair dice are thrown. Consider a bivariate (X, Y). X is 1 if the first die
shows {1 , 3} , 0 otherwise. Y is 1 if the second die shows an even number, 0
otherwise
(a) Determine the probability mass functions (pmfs) for X and Y
(b) Determine the joint pmf of (X, Y)
(c) Determine the pmf of the sum Z = X + Y
(d) Draw the pmf for Z
SE 6.33: The joint pdf of a bivariate r.v. (X, Y) if defined as fXY (x, y) as
K exp [ - (2x + 3y) ], x > 0; y > 0.
(a) Find the value of K
(b) Are X and Y independent?
(c) Are X and Y uncorrelated?
(d) Are X and Y orthogonal?
Hint: ∫ u (dv/dx) dx = u v - ∫ (du/dx) v dx
SE 6.34: Two random variables X and Y are independent with identical uniform
densities U(0, 1). Find the probability density function for the random variable
Z = 4X + Y
11
SE 6.35: Given


 <<<
=
,otherwise,0
,10,
),(
yxk
yxf XY
, determine )|(| yxf YX and ).|(| xyf XY
SE 6.36: An unknown random phase θ is uniformly distributed in the interval
),2,0( π and ,nr +=θ where n ∼ ).,0(
2σN Determine ).|( rf θ
SE 6.37: Let


 <<<
=
.otherwise,0
,1||0,1
),(
xy
yxf XY
. Determine E(X/Y) and E(Y/X)
SE 7.1: Given X(t) = a cos (ωO t + φ ), φ ~ U (0, 2 ∏),
find autocorrelation function RXX (t1, t2).
SE 7.2: Determine the auto correlation function of the random telegraph
signal which may acquire with equal probability at any instant of time either
of the values zero or one, and it makes independent random traversals from
one value to the other
SE 7.3:
SE 7.4: Determine the autocorrelation function of R(t! , t2 ) process of the
SE 7.5; consider the random process X(t) = Y cos ω t , t ≥ 0 where ω is a constant
and Y is a uniform r.v. over (0, 1).
(a) Find E[X(t)
(b) Find autocorrelation function RXX (t1, t2)
(c) Find autocovariance CXX (t1, t2)
12
SE 7.6:
SE 7.7: For the random process X(t) = a cos (ωO t + φ ), φ ~ U (0, 2 ∏) defined in
SE 7.1, determine if it is Strict Sense Stationary and/or Wide Sense Stationary
SE 7.8:
SE 7.9
SE 7.10: (Edge or “change” detector): Let Y(t) = L{X(t)} = X(t) – X(t-1). Find E(Y(t),
RXY(t1, t2) and RYY(t1, t2) .
SE 7.11: Let X(t) be a real valued random process with constant mean function
μX = μ and covariance function CX (t1 , t2 ) = σ2 cos ωO (t1 - t2 ). Determine the
mean and covariance function of the derivative process X’ (t).
SE 7.12: A w.s.s white noise process W(t) is passed through a low pass filter (LPF)
with bandwidth B/2. Find the autocorrelation function of the output process.
13
SE 7.13: Let
1
2
( ) ( )
t T
t TT
Y t X dτ τ
+
−
= ∫
represent a “smoothing” operation using a moving window on the input
process X(t). Find the spectrum of the output Y(t) in term of that of X(t).
SE 7.14: Consider an autocorrelation function that is triangular in shape such that
the correlation goes to zero at shift T.
Determine the power spectral density.
SE 7.15: Two random processes X(t) and Y(t) are given by X(t) = A cos (ωt + ϴ ) and
Y(t) = A sin (ωt + ϴ ) where A and ω are constants and ϴ is a uniform r.v. over
(0, 2 ∏). Find the cross-correlation function of X(t) and Y(t).
SE 7.16
SE 7.17: Find the power spectral density for the exponential autocorrelation
function R (τ) = exp (-α | τ |) with parameter α > 0.
SE 7.18: If autocorrelation R[m] = a|m| , find the power spectral
density function.
SE 7.19: Consider the random process Y(t) = (-1) X(t) where
X(t) is a Poisson process with rate λ.
(a) Find the mean of Y(t)
(b) Find the autocorrelation function of Y(t)
14
SE 7.20: A random process X(t) is created by mixing 2/3rd fair coins (head and tail)
and 1/3rd biased coins (heads on both sides) into boxes. Use rv H = 1, T = 0. Find
the (i) time average and the (ii) ensemble average. (iii) Is the process stationary?
(iv) Is the process ergodic? Hint: For time average, once a coin is chosen, it is used
for all times whereas for across an ensemble for a given time, there is associated
probabilities for choosing the coins.
SE 8.1: 200 phones of type A, 300 of type B, and 100 of type C are mixed together.
Type A are known to be defective with probability 0.03, type B with probability
0.05, and type C with probability 0.1.
(a) If all 600 phones are mixed together and one is selected at random, what is
the probability of it being defective?
(b) Given that the selected chip is part (a) is defective, what is the probability that
it is of type A?
(c) In part (a) after a phone is selected and it is found to be defective, what is the
probability that the next one selected will also be defective.
SE 8.2: The Probability Distribution Function (PDF) if a random variable is
sketched in the figure.
(a) Determine pdf fX (x) and sketch it
(b) Determine probability P [ 0 < x ≤ 2]
(c) Determine the probability P [ 0 ≤ x < 2]
(d) Probability P [ X = 1.5]
(e) Probability P [ X > 2]
SE 8.3: A random variable X is defined to be U (-2, 3). A variable Y is a piecewise
function of X as shown in the figure below.
15
(a) Find and plot the pdf fY (y)
(b) Find and plot PDF FY (y)