53. If U is uniform on (0, 2) and Z, independent of U, is exponential with ...

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

$$

X = \sqrt{2Z}cosU

$$

$$

Y = \sqrt{2Z}sinU

$$

are independent unit normal random variables.