# Integrate an Inequality and Find a Density Function

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.

See attachment for formatted version.

© BrainMass Inc. brainmass.com June 3, 2020, 7:17 pm ad1c9bdddfhttps://brainmass.com/statistics/probability/integrate-inequality-density-function-92338

#### Solution Preview

Please see the attachment. Thank you for using BrainMass.

1. Proof:

We consider the inequality . We integral them on the ...

#### Solution Summary

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.

$2.19