Let X and Y be independent standard normal random variables. Let V = X 2 + Y
Question:
Let X and Y be independent standard normal random variables. Let V = X2 + Y2 and W = tan−1(Y/X).
(a) Show that V and W are independent with V ∼ Exp(1/2) and W ∼ Unif(0, 2π). (x = √vcosw and y = √vsinw.)
(b) Show that this gives the following method for simulating a pair of standard normal random variables given a pair of independent uniforms. Let U1, U2 ∼ Unif(0, 1). Let
X = √−2lnU1 cos(2πU2), Y = √−2lnU1sin(2πU2).
(c) Implement this method and plot 1000 pairs of points.
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a As per the hint x v cos w and y v sin w The Jacobian is The joint density of V and W i...View the full answer
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