QA4 - Multivariate Random Variables

Question 1

Compute the marginal and conditional distributions of a discrete bivariate random variable

Answer 1

Marginal is sum across the row / column
Conditional is divide prob of realisation / prob of overall realisation

Question 2

Define covariance and explain what it measures

Answer 2

Is a measure of dispersion and describes how variables move together

Question 3

Explain the relationship between covariance and correlation of two random variables and how these are related to the independence of the two variables

Answer 3

Two variables must have zero correlation but not necessarily independent if zero correlation

Covariance of two independent variables is also zero

Question 4

Explain the effects of a linear transformation on the covariance and correlation between two random variables

Answer 4

Covariance scales by multipliers

Correlation changes by ab / (mod(a) * mod(b)) where a and b are multipliers

Question 5

Compute the variance of a weighted sum of two random variables

Answer 5

a^2 * Var(X1) + b^2 * Var(X) + 2ab * Cov(X1, X2)