Digital Assignment - 1
ECE2005 – Probability, Theory and Random Process
1. Random variables X and Y are distributed according to the joint PDF
(a) Evaluate the constant a.
(b) Determine the marginal PDF fY (y).
(c) Determine the conditional expectation of 1/X given that Y = 3/2.
(d) Random variable Z is defined by Z = Y − X. Determine the PDF fZ(z).
2. Let X and Y be two independent random variables. Their probability densities functions are
shown below.
1
fX (x) = 34(1 − x2)
fX (x)
0.8
0.6
0.4
0.2
0
−1.5
−1
−0.5
0
0.5
x
1
1.5
fY (y)
1
0.8
2
3
0.6
1
3
0.4
0.2
0
0
Let Z = X + Y . Determine fZ(z).
0.5
1
1.5
y
2
2.5
3
3. Random variables X and Y have the joint PDF shown below:
y
2.0
fX,Y (x,y) = 0.1
1.0
1.0
-1.0
2.0
x
-1.0
-2.0
(a) Find the conditional PDFs fY |X (y | x) and fX|Y (x | y), for various values of x and y,
respectively.
(b) Find E[X | Y = y], E[X], and var(X | Y = y). Use these to calculate var(X).
(c) Find E[Y | X = x], E[Y ], and var(Y | X = x). Use these to calculate var(Y).
4. X and Y are continuous random variables. X takes on values between 0 and 2 while Y takes on
values between 0 and 1. Their joint pdf is indicated below.
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
x
1.2
1.4
1.6
1.8
2
(a) Are X and Y independent? Present a convincing argument for your answer.
(b) Prepare neat, fully labelled plots for fX (x), fY |X (y | 0.5), and fX|Y (x | 0.5).
(c) Let R = XY and let A be the event X < 0.5. Evaluate E[R | A].
(d) Let W = Y − X and determine the cumulative distribution function (CDF) of W .
5. A defective coin minting machine produces coins whose probability of heads is a random variable
P with PDF
In essence, a specific coin produced by this machine will have a fixed probability P = p of giving
heads, but you do not know initially what that probability is. A coin produced by this machine
is selected and tossed repeatedly, with successive tosses assumed independent
Find the probability that the first coin toss results in heads.
(a) Given that the first coin toss resulted in heads, find the conditional PDF of P .
(b) Given that the first coin toss resulted in heads, find the conditional probability of heads on
the second toss.
6. The wombat club has N members, where N is a random variable with PMF
pN (n) = pn−1 (1 − p)
for n = 1, 2, 3, . . ..
On the second Tuesday night of every month, the club holds a meeting. Each wombat member
attends the meeting with probability q, independently of all the other members. If a wombat
attends the meeting, then it brings an amount of money, M , which is a continuous random
variable with PDF
fM (m) = λe−λm for m ≥ 0.
N , M , and whether each wombat member attends are all independent. Determine:
(a) The expectation and variance of the number of wombats showing up to the meeting.
(b) The expectation and variance for the total amount of money brought to the meeting.