05 Normal Distribution: Theory of the Multinormal

Question 1

If X ~ Nₚ(μ, Σ), how is AX + c distributed?

Note: A = q x p matrix, C q x 1 vector

Answer 1

AX + c ~ Nᵩ(Aμ+c, AΣAᵀ)

Question 2

What is Hotelling’s T² distribution?

Answer 2

1) Generalizations of the t-distribution to more parameters
2) multivariate counterpart of the T-test -> “Multivariate” means that you have data for more than one parameter for each sample

Question 3

When is Hotelling’s T² distribution the same as a t-distribution?

Answer 3

When p = 1

Question 4

What is the Wishart distribution?

Answer 4

• Generalization of the univariate chi-square distribution to two or more variables

Question 5

How do you go about solving a question for linear combinations? [e.g. X ~ Nᵢ ( μᵢ, Σ), how is x₁ distributed?]

Answer 5

1) find the matrix/vector(s) to transform X with to get the desired result (e.g. A and x)
2) apply the transformation to μ
3) calculate AΣAᵀ
4) write down result as x₁ ~ N₁(… , …. )

Question 6

How can you find x₂.₃ of some given distribution?

Answer 6

• 2.3 means you want 2 independent of 3
  1) you insert the 2 and 3 into the small letters of the independence formula.
  2) you find the linear transformation to get the resulting formula out of X
  3) you transform the distribution accordingly

Question 7

How to solve a conditional distribution like “find the distribution of (X₁ | X₂ = x₂)”?

Answer 7

• this means X₁ conditional on X₂
  1) you use the formula for conditional distributions
  2) You should get a result like:
  (X₁ | X₂ = x₂) ~ N₁ ( ‘f(x₂)’ , ‘number’ )

Question 8

What is E[M] when M ~ Wₚ (Σ, n)?

Answer 8

E[M] = n • Σ