- A random vector, X, with values in ℝ
^{p}has a multivariate normal distribution written as X ∼N_{p}if for all Borel E ⊆ ℝ^{p},Here Σ is a positive definite symmetric matrix. Recall that λ

_{X}≡ P. Using the change of variables formula, show that λ_{X}defined above is a probability measure. One thing you must show is thatHint: To do this, you might use the fact from linear algebra that Σ = Q

^{∗}DQ where D is a diagonal matrix and Q is an orthogonal matrix. Thus Σ^{−1}= Q^{∗}D^{−1}Q. Maybe you could first letand change the variables. Note that the change of variables formula works fine when the open sets are all of ℝ

^{p}. You don’t need to confine your attention to finite open sets which would be the case with Riemann integrals which are only defined on bounded sets. - Consider the surface z = x
^{2}for∈×. Find the area of this surface. Hint: You can make do with just one chart in this case. Let R^{−1}=^{T},∈×. ThenThen DR

^{−1∗}DR^{−1}=. - A parametrization for most of the sphere of radius a > 0 in three dimensions
is
where we will let ϕ ∈

,θ ∈so there is just one chart involved. As mentioned earlier, this includes all of the sphere except for the line of longitude corresponding to θ = 0. Find a formula for the area of this sphere. Again, we are making do with a single chart. - This problem gives a more elegant treatment of polar coordinates in n dimensions. Let
K⊆Pconsist of all sets of the form E × (a,b] where E is a Borel set in S
^{n−1}, the unit sphere in ℝ^{n},. Hereis the Euclidean norm.- Let θ : S
^{n−1}×→ ℝ^{n}∖be defined by θ≡ ρw. Here w is a unit vector so it is on S^{n−1}. Show that θ is one to one, onto, and continuous with continuous inverse. Here you can let the distance on S^{n−1}×be given byThus θ is a homeomorphism. Explain why θ maps ℬ

one to one and onto ℬ.If ŵ is close to w and

is close to ρ, then ρw is close toŵ and so θ is obviously continuous. Is it one to one? If ρw =ŵ then ρ =and so, cancelling these, you get w = ŵ. Thus it is one to one. Is it onto? Let> 0 then x == θso θ is clearly onto. Is θ^{−1}continuous? Suppose x is close to. Thenis close to. Also=and this is clearly small ifis close enough to x. Therefore, θ^{−1}≡is clearly continuous. It follows that θ maps open sets to open sets and if you definethen G contains the open sets of ℬ

and if B_{i}∈G, then θ= ∪_{i=1}^{∞}θ∈ℬand so G is closed with respect to countable unions. If B ∈G, what about B^{C}?and so

and so, since θ is one to one, the two sets on the right are disjoint. Hence

and the left side is in ℬ

so G is closed with respect to complements. Hence G = ℬ. Similarly θ^{−1}maps Borel sets to Borel sets. This is not surprising because θ is a homeomorphism and considerations about whether a set is Borel come down to considerations about the topology. - Now define a measure on ℬas follows.
Explain why this is a finite measure on ℬ

. - For E × (a,b] ∈K, explain the following computation.
- Let G be those sets F of ℬsuch that
Thus it was shown above that G contains K which is a π system. Explain why. Now explain why G is closed with respect to complements and countable disjoint unions. Explain next why G contains σ

= ℬ. Then explain why for every F ∈ℬconclude that for all E Borel in ℝ

^{n}∖θ

^{−1}=. Thus X_{θ−1(E) }= X_{E}. Thus the integral on the left is of the form - Now explain why for every f ≥ 0 and Borel measurable,

- Let θ : S
- Let V be such that the divergence theorem holds. Show that ∫
_{V }∇⋅dV = ∫_{∂V }udA where n is the exterior normal anddenotes the directional derivative of v in the direction n. Remember the directional derivative.

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