1. Prove that for all x > 0
(1/x - 1/x^1)phi(x) < 1 - ohi(x) < (1/x)phi(x)
Hint: Integrate the following inequalities: (1 - 3y^-4)e^(y^2/2) < e^-(y^2)/2 < (1 + y^-2)e^-(y^2)/2.
2. Let X be a random number from (0,1) (ie. X ~ uniform (0,1)). Find the density function of (a) Y _ -log(1-X), (b) Z _ X^n.
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This solution provides a proof for all x>0 by integration the inequalities (1 - 3y^-4)e^(y^2/2) < e^-(y^2)/2 < (1 + y^-2)e^-(y^2)/2 and also calculates the density function of log functions. All steps are shown in a stepwise manner.