Let X = (X₁, X₂) be a two-dimensional normal random vector; E{X₁} = 1, E{X²₁} = 5, E{X₂} = 2, E{X²₁} = 13, E{X₁X₂} = −2.

(a) Write the density of the centered vector Y = X−m, where the vector m = (1,2). Write the density of X.

(b) For t = (3,−4), (i) write the expectation, the variance and the density of t₁X₁ +t₂X₂; (ii) write the expectation, the variance and the density of the projection of X on t.