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June 4 Lecture
Random Vector Generation
We want to sample from X=(X1,X2,...,Xd),a d-dimensional vector.

Case 1:
If X1, X2, ..., Xd are all independent, then
we can sample from eachcomponent X1, X2,..., Xd individually and then form a vector.
Since f(x)= f(x1, x2, ..., xd)= f(x1)*f(x2)*...*f(xd).

Case 2:
If X1, X2, ..., Xd are not independent, then
we need to know the conditional distribution of f(x2|x1), f(x3|x2,x1),..., f(xd|x(d-1),...,x1).
Since f(x)= f(x2|x1)*f(x3|x2,x1)*...* f(xd|x(d-1),...,x1).

Example:generate Uniform random vector
X= (x1,..., sd) from the d-dimensional rectangle
D= {(x1,..., sd): ai <= bi, i= 1,..., d}


Code:

for i= 1,..., d
  Ui~U(0, 1)
  Xi= ai + U(bi-ai)
end
  X=(x1,..., xd)