Question 1
Typing speed (TS), in words per minute, can be used as a behavioural biometric to enable
identification of computer users in cases of abuse. A system manager reports that a user
account has been breached i.e. has been accessed by someone other than it's owner. It
has been possible to restrict the possible abusers to two people, denoted as A and B. It
is known that the average typing speeds of these users can be approximately modelled as
Gaussians with parameters (mean, standard deviation) of (70,6), (60,3) respectively. It is
also known that user A visits the computer labs twice as often as B. The typing speed of
the abuser is measured as 64 words per minute.
(a) plot the following: P(TS/A), P(TS/B), P(A/TS) and P(B/TS)
(b) the computer manager adopts a decision rule such that if typing speed is less than 65
words per minute the culprit must be B, while if equal to or above 65 then it must
be A i.e. identifies abuser via closest mean. What is the error probability associated
with this decision rule i.e. how often will this rule lead to an incorrect decision?
Question 2
In a simple binary communication channel 1s and 0 s are transmitted and received. How-
ever, communication channel noise can cause signal corruption. Define events T0 and T1
as the transmission of a 0 or a 1 respectively and events R0 and R1 as reception of a 0
or a 1 respectively. The noise aspects of the communication channel can be modelled by
the following conditional probabilities
P(R1/T1) = 0.95, P(R0/T1) = 0.05, P(R1/T0) = 0.05, P(R0/T0) = 0.95
Given that the prior transmission probabilities for each of the symbols are P(T1) =
0.65 and P(T0) = 0.35 calculate
(a) the posteriori reception probabilities P(R1) and P(R0)
(b) the probabilities of system error, P(T1/R0) and P(T0/R1)
Question 3
A computer chip manufacturer finds that for every 100 chips produced, 85 meet specifica-
tions, 10 need reworking and 5 need to be discarded. Ten chips are chosen for inspection.
(a) What is the probability that all 10 meet specs?
(b) What is the probability that two or more need to be discarded?
(c) What is the probability that 8 meet specs, 1 needs rework and 1 will be discarded?
Question 4
Consider a pair of random variables, X and Y, that are jointly uniformly distributed over
the unit circle i.e. the joint pdf is given by
pXY (x; y) ={c x2 + y2 < 1
0 otherwise
(a) calculate a value for the constant c.
(b) derive an analytic formula for the marginal probabilities, pX(x) and pY (y).
(c) are the variables X and Y independent?
