User:Egm6341.s11.team4.KTH/HW5

=Problem 5.3 Cauchy Heavy Tails=

Given
For 1) and 2), Quartile points: $$ Q_1 \ $$, $$ Q_2 \ $$ and $$ Q_3 \ $$.

Cauchy pdf :

Gauss(normal) pdf :

For 3) and 4) $$ x_0 = \mu = 0 \ $$ and $$ \gamma^{C}=1 \ $$ $$ \gamma^{C}= \ $$ half width of $$ C(x_0, \gamma^{C}) \ $$ $$ \gamma^{G}= \ $$ half width of $$ N(\mu, \gamma^{C}) \ $$

Objectives
1) Find $$ \left \{ Q_1, Q_2 \right \} \ $$ for $$ C \left ( x_0, \gamma \right ) \ $$ 2) Find $$ \left \{ Q_1, Q_2 \right \} \ $$ for $$ N \left ( \mu, \sigma \right ) \ $$ 3) Find $$ \sigma\ ^1$$ such that $$ \gamma\ ^G =1 $$, and Plot $$ C \left ( 0,1 \right ) $$ and $$ N \left ( 0, \sigma\ ^1 \right ) $$ 4) Find and plot $$ \left \{ Q_1^c, Q_3^c \right \}  $$ for $$ C \left ( x_0, \gamma\ \right ) $$ and $$ C \left ( 0, 1 \right ) $$ also find and plot $$ \left \{ Q_1^G,Q_3^G  \right \}  $$ for $$ N \left ( \mu\, \sigma\ \right ) $$ and $$ N \left ( 0, \sigma\ ^1 \right ) $$

1) The cumulative distribution function of Cauchy pdf is
Substitue Q1 into x of eq(3.1) we get eq(3.2).

From eq(3.2)

Substitue Q3 into x of eq(3.1) we get eq(3.4).

From eq(3.5)

2) The cumulative distribution function of Gauss pdf is
Substitue Q1 into x of eq(3.8) we get eq(3.8).

Substitue Q3 into x of eq(3.8) we get eq(3.10).

3) Find standard deviation for half width of Gaussian distribution
Gaussian pdf is

The height of Gaussian pdf is

We want to find sigma which will place the half width at 1.

MatLab Code

Plot

===4) Find $$ \left \{ Q_{1}^{C}, Q_{3}^{C} \right \} $$ for $$ C(x_0,\gamma) \ $$ and $$ C(0,1) \ $$ and find $$ \left \{ Q_{1}^{G}, Q_{3}^{G} \right \} $$ for $$ N(\mu,\sigma) \ $$ and $$ N(0,\sigma^1) \ $$.===

From 1)

From 1)

Plot $$ \left \{ Q_{1}^{C}, Q_{3}^{C} \right \} $$ for $$ C(0,1) \ $$ and $$ \left \{ Q_{1}^{G}, Q_{3}^{G} \right \} $$ for $$ N(0,\sigma^1) \ $$

MatLab Code

Plot