Chapter 2
Introduction to probability
Problem 2.1 Give a real-world example of a joint distribution P r(x, y) where x is discrete
and y is continuous.
Problem 2.2 What remains if I marginalize a joint distribution P r(v, w, x, y, z) over five
variables with respect to variables w and y? What remains if I marginalize the resulting
distribution with respect to v?
Problem 2.3 Show that the following relation is true:
P r(w, x, y, z) = P r(x, y)P r(z|w, x, y)P r(w|x, y).
Problem 2.4 In my pocket there are two coins. Coin 1 is unbiased, so the likelihood
P r(h = 1|c = 1) of getting heads is 0.5 and the likelihood P r(h = 0|c = 1) of getting
tails is also 0.5. Coin 2 is biased, so the likelihood P r(h = 1|c = 2) of getting heads is
0.8 and the likelihood P r(h = 0|c = 2) of getting tails is 0.2. I reach into my pocket and
draw one of the coins at random. There is an equal prior probability I might have picked
either coin. I flip the coin and observe a head. Use Bayes’ rule to compute the posterior
probability that I chose coin 2.
Problem 2.5 If variables x and y are independent and variables x and z are independent,
does it follow that variables y and z are independent?
Answer
No, it does not follow. Consider any general distribution P r(y, z) where y and z are NOT
independent. Now consider the marginal distributions P r(y) and P r(z). It is perfectly
possible to have a third distribution P r(x) which does not provide any information about
y or z and hence is independent of each and P r(x, y) = P r(x)P r(y) and P r(x, z) =
P r(x)P r(z).
If you are unsure about this, then construct a counter example where x, y and z are
all discrete variables with two entries. Construct a non-independent 2 × 2 distribution
between y and z, marginalize it with respect to each and then construct 2×2 independent
joint distributions between x and z and x and y.
Copyright
c
2012 by Simon Prince. This latest version of this document can be downloaded from
http://www.computervisionmodels.com.
