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If U is uniform on 0 2 and Z independent

If U is uniform on (0, 2π) and Z, independent of U, is exponential with rate 1, show directly (without using the results of Example 7b) that X and Y defined by

X = √2ZcosU

Y = √2ZsinU

are independent standard normal random variables.

X = √2ZcosU

Y = √2ZsinU

are independent standard normal random variables.

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