Statistics for …nance
a.a. 2021/22
REVIEW EXERCISES
Exercise 1.
Let’s consider the events A and B (associated to the same random experiment) that are incompatible
and such that:
P(A) = 0:2
P(B) = 0:4:
Compute the probability of the following events:
1. A [ B;
2. A \ B,
3. A \ B,
4. A \ (A \ B),
5. A [ (A \ B).
[0.8, 0.2, 1, 0.2, 1]
Exercise 2.
Let’s consider the events A and B associated to the same random experiments and such that:
P(B) = 0:5
P(A \ B) = 0:2
P(BjA) = 0:4:
Compute the following probabilities and provide a reasoned answer to the questions posed.
1. P(A)
2. P(A [ B)
3. P(AjB)
4. Are the events A and B incompatible?
5. Are the events A and B independent?
[0.5, 0.8, 0.4]
Exercise 3.
The events A and B, associated to a given random experiment, are independent and such that:
P (A \ B) = 0:2
P (A [ B) = 0:7
Compute P (A) and P (B).
[0.4, 0.5]
1
P (B) > P (A) > 0
Exercise 4.
The events A and B (associated to the same sample space) are incompatible. One knows that:
P (A) = 0:2
P AjB = 0:5
By using the de…nitions and the Venn diagrams, compute:
1. P (B)
2. P A \ B .
[0.6, 0.2]
Exercise 5.
Let’s consider the events A and B (associated to the same sample space) and suppose that:
P (A) = 0:6
P (BjA) = 0:7
P BjA = 0:2:
1. Compute P (A \ B) and P B \ A .
2. With the help of a Venn diagram which allows to visualize the relationship between A \ B and
B \ A, compute P (B).
[0.42, 0.08, 0.5]
PROBLEMS
Problem 1 - False positive problem
A credit risk diagnostic system of a given bank classi…es the …rms in bonis or default according to some
economical aspects obtained by the …nancial statements of the same. That system presents the following
features:
the probability that the system classi…es a …rm in bonis when it is actually in bonis equals to 0:85;
the probability that the system classi…es a …rm in default when it is equals to 0:90.
Knowing that the default probability is 7%, answer the following questions:
1. What is the probability that the system produces a false positive, that is it classi…es in default a
healthy business? [0.15]
2. If a …rm is classi…ed in default by the system, what is the probability that it is? [0.311] The latter
result is surprisingly large: comment on it according to the used probabilities.
2