Let X and Y be independent standard normal random variables Let V X 2 Y 2 and W tan1(Y X) (a) Show that V and W are independent with V Exp(1 2) and W Unif(0, 2) (x vcosw and y vsinw ) (b) Show that this gives the following method for simulating a pair of standard normal random variables given a pai...

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 ...

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).

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