Exercise 11.7 (Box–M¨uller) For independent random variables U1 and U2 that follow U(0, 1), define

Prove that E[Z1] = E[Z2] = 0, V[Z1] = V[Z2] = 1 and E[Z1Z2] = 0. Note: It is known that Z1 and Z2 are independent and follow N(0, 1).

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Z =-2log U sin(2U), Z = -2log U cos(2U2).