QTA 4 - MULTIVARIATE RANDOM VARIABLES

Question 1

How can a probability matrix be used in relation to a probability mass function (PMF)?

Answer 1

A probability matrix relates realizations to probabilities and serves as a tabular representation of a PMF

It describes discrete distributions defined over a finite set of values.

Question 2

What does the PMF of a bivariate random variable represent?

Answer 2

The PMF returns the probability that two random variables each take a certain value

It requires three axes: X1, X2, and the probability mass/density.

Question 3

Define covariance.

Answer 3

Covariance is a measure of how two random variables move together.

Question 4

What does the expectation of a function for a bivariate discrete random variable represent?

Answer 4

It is a probability weighted average of the function of the outcomes.

Question 5

How is the marginal PMF of a bivariate random variable computed?

Answer 5

It is computed by summing the joint PMF across all values of the other variable.

Question 6

What is a marginal distribution?

Answer 6

The distribution of a single component of a bivariate random variable.

Question 7

What condition must be met for two random variables to be independent?

Answer 7

The joint PMF must equal the product of the marginal PMFs.

Question 8

What is the relationship between covariance and correlation?

Answer 8

Covariance measures the direction of the relationship, while correlation measures the strength and direction of the relationship.

Question 9

What is the formula for the PMF of a trinomial random variable?

Answer 9

fX1,X2 = (n! / (x1! x2! (n - x1 - x2)!)) * p1^x1 * p2^x2 * (1 - p1 - p2)^(n - x1 - x2)

Question 10

What is the definition of a conditional distribution?

Answer 10

It summarizes the probability of outcomes for one random variable given that another takes a specific value.

Question 11

Fill in the blank: The expectation of a function g(X1, X2) is defined as E[g(X1, X2)] = ___

Answer 11

ΣΣ g(x1, x2)fX1,X2

Question 12

What does the CDF of a bivariate variable return?

Answer 12

It returns the total probability that each component is less than or equal to a given value.

Question 13

True or False: The components of a bivariate random variable are independent if the joint PMF is equal to the sum of the marginal PMFs.

Answer 13

False

Question 14

What is the significance of the i.i.d property in random variables?

Answer 14

It is helpful in computing the mean and variance of a sum of i.i.d random variables.

Question 15

How is the variance of a weighted sum of two random variables computed?

Answer 15

It involves the variances and covariances of the random variables.

Question 16

What are the two components of a bivariate random variable?

Answer 16

X1 and X2.

Question 17

What is the relationship between the marginal PMF and the marginal CDF?

Answer 17

The marginal CDF is defined using the marginal PMF to measure the total probability less than a given value.

Question 18

What does the first moment of X represent?

Answer 18

The mean E[X] = [μ1, μ2].

Question 19

What is the definition of the conditional probability of two events?

Answer 19

P(A|B) = P(A ∩ B) / P(B).

Question 20

True or False: Knowledge about the value of X2 must contain information about X1 for independence.

Answer 20

False

Question 21

What is the formula for covariance between X1 and X2?

Answer 21

Cov[X1, X2] = E[(X1 - E[X1])(X2 - E[X2])] = E[X1X2] - E[X1]E[X2].

Question 22

How is the conditional PMF computed for a bivariate random variable?

Answer 22

It is the joint probability divided by the marginal probability of the conditioning variable.

Question 23

What is the definition of covariance?

Answer 23

Covariance is a measure of dispersion that captures how the variables move together.

Question 24

How is the covariance between two variables (X_1) and (X_2) defined?

Answer 24

Cov[X1, X2] = E[(X1 - E[X1])(X2 - E[X2])] = E[X1X2] - E[X1]E[X2]

Question 25

What does the covariance of a variable with itself represent?

Answer 25

The covariance of a variable with itself is just the variance of that variable.

Question 26

In a bivariate random variable, how many variances and covariances are there?

Answer 26

There are two variances and one covariance.

Question 27

What is the common abbreviation for the variance of (X_1) and (X_2)?

Answer 27

V[X1] is denoted as σ² and V[X2] as σ².

Question 28

What does the symbol σ12 represent?

Answer 28

The covariance between (X_1) and (X_2).

Question 29

What is the range of values that covariance can take?

Answer 29

Covariance can take on values from -∞ to +∞.

Question 30

Why is correlation often reported instead of covariance?

Answer 30

Correlation is a scale-free measure and is easier to interpret.

Question 31

How is correlation between two variables obtained?

Answer 31

By dividing their covariance by their respective standard deviations.

Question 32

What does correlation measure?

Answer 32

The strength of the linear relationship between two variables.

Question 33

What is the range of correlation values?

Answer 33

Correlation is always between -1 and 1.

Question 34

What does a positive correlation indicate?

Answer 34

When (X_1) and (X_2) tend to increase together.

Question 35

What does a negative correlation indicate?

Answer 35

If (X_2) tends to decrease when (X_1) increases.

Question 36

How do location shifts affect covariance?

Answer 36

Location shifts have no effect on covariance.

Question 37

How does scaling affect covariance?

Answer 37

The scale of each component contributes multiplicatively to the change in covariance.

Question 38

What is the formula for covariance after applying shifts and rescaling?

Answer 38

Cov[a + bX1, c + dX2] = bd Cov[X1, X2]

Question 39

What is the relationship between correlation and covariance when scaling?

Answer 39

Corr[aX1, bX2] = sign(a) sign(b) Corr[X1, X2]

Question 40

When is the covariance between two independent random variables zero?

Answer 40

Cov[X1, X2] = 0.

Question 41

What is the relationship between covariance and variance of a random variable?

Answer 41

Cov[X1, X1] = Var(X1).

Question 42

What are the variance formulas when (X_1) and (X_2) are not independent?

Answer 42

V[X1 + X2] = V[X1] + V[X2] + 2 Cov[X1, X2] and V[X1 - X2] = V[X1] + V[X2] - 2 Cov[X1, X2].

Question 43

What does the correlation coefficient (ρ) indicate?

Answer 43

The strength and direction of a linear relationship between two random variables.

Question 44

What does a correlation of ρ = -1 indicate?

Answer 44

A perfect negative relationship.

Question 45

What does a correlation of ρ = 0 indicate?

Answer 45

No linear relationship.

Question 46

What does a correlation of ρ = 1 indicate?

Answer 46

A perfect positive relationship.

Question 47

What happens to correlation when two random variables are independent?

Answer 47

They must have zero correlation.

Question 48

Can two variables have zero correlation and still be dependent?

Answer 48

Yes, they can be dependent without having a linear relationship.

Question 49

What is coskewness?

Answer 49

The third cross central moment indicating simultaneous extreme deviations of two random variables.

Question 50

What is cokurtosis?

Answer 50

The fourth cross central moment indicating simultaneous extreme positive and negative deviations of two random variables.

Question 51

What is a conditional expectation?

Answer 51

An expectation when one random variable takes a specific value or falls into a defined range.

Question 52

What is a conditional variance?

Answer 52

The variance of a random variable given another variable.

Question 53

What can cause dependence between random variables in finance?

Answer 53

Shifts in investor risk aversion, cross-asset spillovers, crowded portfolio strategies.

Question 54

What is the role of conditioning in random variables?

Answer 54

Conditioning helps to remove the dependence between variables.

Question 55

How does the transition from discrete to continuous random variables affect calculations?

Answer 55

It involves replacing PMF with PDF and sums with integrals.

Question 56

What is the form of a probability density function (PDF) for continuous random variables?

Answer 56

The PDF has the same form as a PMF and is written as 𝑓X1,X2(𝑥1, 𝑥2)

Question 57

How is the probability in a rectangular region defined for continuous random variables?

Answer 57

Pr(𝑙1 < 𝑋1 < 𝑢1 ∩ 𝑙2 < 𝑋2 < 𝑢2) = ƒ ƒ 𝑓X1,X2(𝑥1, 𝑥2) 𝑑𝑥1𝑑𝑥2

Question 58

What does the cumulative distribution function (CDF) represent in the context of continuous random variables?

Answer 58

The CDF is the area of a rectangular region under the PDF where the lower bounds are —∞ and the upper bounds are the arguments in the CDF.

Question 59

What must a PDF function always do?

Answer 59

A PDF function is always non-negative and must integrate to one across the support of the two components.

Question 60

What is the simplest form of continuous multivariate random variable?

Answer 60

The standard bivariate uniform where the two components are independent.

Question 61

What is the PDF of the standard bivariate independent uniform random variable?

Answer 61

𝑓X1,X2 = 𝑘, where the support of the density is the unit square.

Question 62

What is the CDF of the standard bivariate independent uniform random variable?

Answer 62

𝐹X1,X2 = 𝑥1𝑥2

Question 63

True or False: The transition from discrete multivariate random variables to continuous ones primarily changes the PMF to a PDF.

Answer 63

True

Question 64

Fill in the blank: The most substantial change when moving from discrete to continuous random variables is the switch from sums to _______.

Answer 64

integrals

Question 65

What must be adjusted if the PDF is not defined for all values?

Answer 65

The lower bounds can be adjusted to be the smallest values where the random variable has support.

Question 66

What denotes conditional independence in the context of distributions?

Answer 66

This distribution is therefore conditionally independent even though the original distribution is not.