Inference Using Multivariate Normal Distribution

Question 1

Give the pdf of a MN distribution

Answer 1

1/[2π^(p/2) |Σ|^1/2] ) * exp( -0.5(x_-mu_)^T Σ^(-1) (x_-mu_) )

Question 2

How many parameters are there in a multivariate normal distribution?

Answer 2

1/2 * p * (p+3)

Question 3

Give the marginal distribution of x1

Answer 3

1/[2π^(k/2) |Σ_11|^1/2] ) * exp( -0.5(x1_-mu1)^T Σ^(-1) (x_1-mu1) )

Question 4

Give the conditional distribution of x1 given x2

Answer 4

X1|X2 ~ MN ( mu1_ + Σ12Σ22^(-1)(x2_ - mu_) , Σ22 - Σ21Σ11^(-1) Σ12

Question 5

What does the partial correlation coefficient measure?

Answer 5

Strength of dependence between two variables given values of other variables

Question 6

Give the formula for ρ12|3

Answer 6

(ρ12 - ρ13 * ρ23) / sqrt ( (1-ρ13^2) * (1-ρ23^2) )

Question 7

How do you estimate ρ12|3 in terms of regression?

Answer 7

Corr ( X1 - a1X3, X2 - a2X3) = ρ12|3

a1 = σ13 / σ3^2
a2 = σ12 / σ3^2

Question 8

Give the test statistic for a one-sample Σ known test

Answer 8

H0: mu_ = mu0_

• 2*log(λ) = n(xbar_ - mu0)^T * Σ^(-1) (xbar - mu0_)

~ X^2_r, r is difference in parameter spaces for H0 and H1

Question 9

Give the test statistic for a one-sample, Σ unknown test

Answer 9

H0: mu_ = mu0_

T^2 = n(xbar_ - mu0)^T * S^(-1) (xbar - mu0_)

Now T^2 * (n-p) / ((n-1)*p) ~ F(p, n-p)

Question 10

Give the test statistic for a two-sample, Σ unknown test

Answer 10

H0: mu1_ = mu2_

Find S = [(n1 - 1)S1 + (n2 - 1)S2] / (n1 + n2 - 2)

T = [n1*n2 / (n1+n2)] * (Xbar_ - Ybar_)^T * S^(-1) * (Xbar_ - Ybar_)

Now ( [n1n2(n1+n2-p-1)]/ [p(n1+n2)(n1+n2-)] ) * T ~ F(p, n1 + n2 - p - 1)