Question

Let X be a multivariate normal random vector N3(μ, Σ) with μ' = [-3, 1, 4] and Σ as shown.

a. What is the conditional distribution of X1, given that X3 = x3?

b. What is the conditional distribution of X1, given that X2 = 2 and X3 = x3?

Answer 1

The student's questions pertain to finding conditional distributions within a multivariate normal random vector, considering specified values for other elements of the vector. These calculations are rooted in the properties of the multivariate normal distribution, involving covariances and adjustment of means.

Step-by-step explanation:

The student is asking about the conditional distributions of a multivariate normal random vector given certain fixed values of other elements of this vector. To be precise, they are interested in the conditional distribution of X1, given X3 = x3, and the conditional distribution of X1, given both X2 = 2 and X3 = x3. The multivariate normal distribution has properties which allow us to compute these conditional distributions using the provided mean vector μ and covariance matrix Σ.

To find the conditional distribution of X1 given X3 = x3, we need to take into account the covariance between X1 and X3 as represented by Σ, and adjust the mean of X1 accordingly. Similarly, to find the conditional distribution of X1 given X2 = 2 and X3 = x3, we perform a similar adjustment taking into account the covariances between X1, X2, and X3.

This requires computation using the formulas for conditional distributions in a multivariate normal setting, often involving the inversion of covariances matrices and manipulation of vector algebra.