Exercises Typology

Question 1

A coin is flipped 100 times. Given that there were 55 heads, find the maximum
likelihood estimate for the probability p of heads on a single toss.

Question 2

Consider these 2 probability distributions on the same space Ω = {A, B, C, D}
p = {1/2, 1/4, 1/8, 1/8}, q = {1/4, 1/8, 1/8, 1/2}.
Compute the KL divergence when p is the true distribution and q its
approximation and in the opposite case

Question 3

On 4 families of 2 members we measure the income of February X and the related
expenses for food Y
X : 1500 1700 1400 1600
Y : 200 350 150 300
Calculate the covariance and correlation coefficient and interpret them.

Question 4

Describe the correlation in the 3 examples represented below: (we have image photos)

Question 5

A game consists of tossing a fair die and a coin. As result of the roll of the die the
number reported on the upper face is considered, while for the coin toss is
considered 0 score if it comes heads, 1 score if shows cross. Determine the
expected value and variance of the random variable which describes the sum of
the scores reported in the toss of the dice and the coin.

Question 6

Why is the covariance matrix of a random vector symmetric?
Definition of the covariance matrix of a random vector.
Describe what is meant by overfitting
Definition of KL divergence
Briefly describe the difference between supervised and unsupervised learning.
Definition of Maximum Likelihood Estimate.
When is Maximum Likelihood Estimate calculated? (theorical)

Question 7

An urn contains three balls marked by the numbers 1, 2, 3.
Remove with repositioning a sample of size two.
Let Y be the random variable expressing the arithmetic mean of the numbers
shown on the drawn balls.
Calculate the variance of Y .

Question 8

The marketing manager of a company wants to establish the effect of the
advertising expenses (in hundreds of euros) on the respective income (in
thousands of Euros).
A sample of 5 local units is extracted and the following results are obtained:

      Advertising              Income

Unit 1 1 1
Unit 2 2 1
Unit 3 3 2
Unit 4 4 2
Unit 5 5 4

Evaluate the covariance for this example, the correlation coefficient, ecc…

Question 9

Describe the correlation in the 3 examples represented below:

Question 10

Consider the following data and compute the correlation coefficient between X
and Y
Company X Stock Value ($)
1.245
1.415
1.312
1.427
1.510
1.590

Company Y Stock Value ($)
100
123
129
143
150
197

Answer 10

r (x, y) (correlation coefficient) = 0.9157

Question 11

On 4 families of 2 members we measure the income of February X and the related
expenses for food Y
X : 1500 1700 1400 1600
Y : 200 350 150 300
Calculate the covariance and correlation coefficient and interpret them.

Answer 11

cov (x,y) = 8750
r (x, y) = 0.989
we know for the correlation coefficient that X and Y as r.v are strictly correlated.

Question 12

Compute the correlation coefficient between X and Y and the covariance

x y
2 4
2 3
4 1
5 1
7 2
8 1
10 0

Answer 12

r (x,y) = -0,7913
cov (x,y) = -2,87

X and Y are strictly correlated in an inverse way, so when X increase Y decrease and viceversa

Question 13

A game consists of tossing a fair die and a coin. As result of the roll of the die the
number reported on the upper face is considered, while for the coin toss is
considered 0 score if it comes heads, 1 score if shows cross. Determine the
expected value and variance of the random variable which describes the sum of
the scores reported in the toss of the dice and the coin.

Question 14

An urn contains three balls marked by the numbers 1, 2, 3.
Remove with repositioning a sample of size two.
Let Y be the random variable expressing the arithmetic mean of the numbers
shown on the drawn balls.
Calculate the variance of Y .

Question 15

Evaluate the covariance and the correlation matrix for this example.

x y z
75 10.5 45
65 12.8 65
22 7.3 74
15 2.1 76
18 9.2 56