Computer Science City College of New
York
CSc 59866 - CAPSTONE I
Spoken Dialog Systems and Voice XML
Assignment 5
Problem 1 Given a two-class one-dimensional classification problem (fish
classification), with 1 representing the class of sea-basses ,  2 the class of salmons,
and x being the measured lightness of the fish, with the following parameters (uniform
distributions):
1  3 / 4  x  1 / 4
1 0  x  1 
p( x |  )  
p( x |  )  


1
2
elsewhere 
0
0 elsewhere
where P(1 )  P(2 )  1 / 2.
(a) Plot the distributions.
(b) Write the Bayes decision rule for this case (hint: draw the decision boundary). Is this
solution unique? Explain.
(c) Compute the probability of error.
(d) How will the decision surface change if the priors are not equal? Explain.
(e) How will the probability of error change if the priors are not equal? Compute the
probability of error for the case P(1 )  0.8P(2 )  0.2.
Problem 2 .
Consider our medical diagnosis example: (note: we went through (a)-(e) in class)
It is known that 1% of population suffers from a particular disease. A blood test has a
97% chance to identify the disease for a diseased individual, by also has a 6% chance of
falsely indicating that a healthy person has a disease.
(a.) What is the probability that a random person has a positive blood test?
(b.) If a blood test is positive, what’s the probability that the person has the disease?
(c.) If a blood test is negative, what’s the probability that the person does not have the
disease?
A physician can do three possible actions after seeing patient’s test results:
 Action A1 : Decide the patient is sick
 Action A2: Decide the patient is healthy
 Action A3: Send the patient for another test
The costs of those actions are:
1. If the patient is healthy, but the doctor decides he/she is sick - $20,000.
2. If the patient is sick, but the doctor decides he/she is healthy - $100.000
3. Sending the patient for another test costs $10,000
What is the minimum-cost decision when
(d) the results of the test are positive
(e) the results of the test are negative
Assume conditional independence of multiple test results, i.e.
P(T , T | S )  P(T | S ) P(T | S ) , where T is a Boolean random variable indicating the
1 2
1
2
i
result of i-th test, i = 1, 2; and S a Boolean random variable indicating the health status
of a person undergoing the tests (has the disease or not).
If the physician decides to send a patient for which the first test result was positive to
another test, what should be the physician’s minimum-cost decision when:
(f) the results of the second test are positive
(g) the results of the second test are negative