Problems
Worksheet Week 21
Q1. X and Y are two (binary) random variables. If X and Y are independent, then P (X, Y ) = P (X)P (Y )
(a) Give an example of two random variables that are independent.
(b) Complete the probability table below in such way that the variables X and Y are
independent.
X=0 X=1 Y=0
Y=1
(c) Determine the missing entries (a, b) of the joint distribution in such a way that the
variables X and Y are again independent.
P(Y =0,X=0)=0.1 P(Y =0,X=1)=0.3 P(Y =1,X=0)=a P(Y =1,X=1)=b
Q2. Consider the following Bayesian network:
2
(a) Which random variables are independent of X3,1?
(b) Which random variables are independent of X3,1 given X1,1?
Q3. Solve the questions on slides 42 and 44 of the lecture slides.
Q4. A patient can have a symptom, S, that is caused by two different diseases, A and B. It is known that the presence of a gene G is important in the manifestation of disease A. The Bayes net and conditional probability tables are shown in Figure 2.
G
AB S
g ¬g
P(G) 0.1 0.9
P(B) b 0.4 ¬b 0.6
P(S A, B)
a b s 1.0 a b ¬s 0.0 a ¬b s 0.9 a ¬b ¬s 0.1 ¬a b s 0.8 ¬a b ¬s 0.2 ¬a ¬b s 0.1 ¬a ¬b ¬s 0.9
P(A G) ga1.0 g¬a0.0 ¬ga0.1 ¬g ¬a 0.9
Figure 1: Bayes net and probability tables for Q5
(a) What is the probability that a patient has disease A
3 turn over/Qu. continues . . .
(b) What is the probability that a patient has disease A if we know that the patient has disease B
(c) What is the probability that a patient has disease A if we know that the patient has disease B AND symptom S
