User:EGM6341.s11.TEAM1.Yoon.HW5

P3-1
Note: Culmulative distribution function(CDF) of Cauchy distribution:

Plugging Q1 into CDF function and value of 1/4 yields,

Plugging Q3 into CDF function and value of 3/4 yields,

p3-2
Note: Cumulative distribution function (CDF):

Plugging x=Q1 into the CDF function,

Multiplying 2 on both side of the eqn,

Subtracting 1 from the both side,

Tanking inverse erf function on both side of the eqn,

Multiplying sqrt(2) and sigma from the both side,

Adding mu on the both side yields,

Similarlly,

Multiplying 2 on both side of the eqn,

Subtracting 1 from the both side,

Tanking inverse erf function on both side of the eqn,

Multiplying sqrt(2) and sigma from the both side,

Adding mu on the both side yields,