04 Random Variables: Multivariate Distributions

Question 1

What is a cumulative distribution function?

Answer 1

a function whose value is the probability that a corresponding continuous random variable has a value less than or equal to the argument of the function:

F (x) = P (X ≤ x ) = P (X₁ ≤ x₁, X₂ ≤ x₂, …. Xₚ ≤ xₚ)

Question 2

What is the probability density function?

Answer 2

1) a function of a continuous random variable

2) whose integral across an interval gives the probability that the value of the variable lies within the same interval

Question 3

What is the marginal probability density function?

Answer 3

1) gives marginal probability of a continuous variable
2) refers to the probability of a particular event taking place without knowing the probability of the other variables = probability of a single variable occurring
3) independent of other variables in the sample space

Question 4

What is the marginal probability of an event X?

Answer 4

anytime you are in interested in a single event irrespective of any other event (i.e. “marginalizing the other event”), then it is a marginal probability.

Question 5

What is conditional probability?

Answer 5

Conditional probability is the probability of one thing being true given that another thing is true, and is the key concept in Bayes’ theorem.

Question 6

What is the conditional probability density function?

Answer 6

• some information gives rise to a conditional probability distribution
• such a distribution can be characterized by a conditional probability density function

Question 7

How can probability distributions help understand whether x₁ and x₂ are independent?

Answer 7

1) x₁ and x₂ are independent if and only if their joined probability consist of two marginal functions
2) that means that when calculating the conditional pdf for x₁ given x₂, only the marginal pdf of x₁ remains

Question 8

What do expectations have to do with probability density functions?

Answer 8

μ₁ = E[x₁] = integral of ( x₁ * marginal probability density function of x₁)

Question 9

How can you write down that X is normally distributed?

Answer 9

X ~ Nₚ ( μ, Σ)

Question 10

What is the Mahalanobis transformation?

Answer 10

1) linear matrix transformation
2) eliminates the correlation between the variables
3) standardizes the variance of each variable.

Result: a standardized, uncorrelated data matrix Z

=> transform X into Y and study how Y is distributed

Question 11

What are the properties of a multivariate mixture model?

Answer 11

• k components with individual densities fₖ()
• individual weights wₖ
• k = 1, …. , K

Question 12

-∞∫∞ xᵏ dx = ?

Answer 12

xᵏ⁺¹ / (k + 1)

Question 13

-∞∫∞ x² dx = ?

Answer 13

x³/ 3

Question 14

-∞∫∞ dx = ?

Answer 14

-∞∫∞ 1 dx
= -∞∫∞ x° dx
= x

Question 15

How can you show that f(•) is a density function of X?

Answer 15

1) show that the integral -∞∫∞ dx is equal to 1
example:
₀∫¹ ₀∫¹ (•) dx₁ dx₂

Question 16

when evaluating a multivariate normal distribution function at f(0ₚ), what is the result?

Answer 16

the exponential vanishes (because e⁰ = 1), so that only the term |2πΣ| remains