User:Egm6341.s11.team2.zou/HW5

=Problem 5.3 Cauchy Distribution and Normal Distribution=

From the lecture slide Mtg26

Given:
The Cauchy distribution:

The Normal or Gauss distribution:

and the definition of quartiles $$\displaystyle {{Q}_{1}}$$, $$\displaystyle {{Q}_{2}}$$ and $$\displaystyle {{Q}_{3}}$$:

Find:
a) Three quartiles $$\displaystyle \left\{ Q_{1}^{C},Q_{3}^{C} \right\}$$ for Cauchy distribution

b) Three quartiles $$\displaystyle \left\{ Q_{1}^{G},Q_{3}^{G} \right\}$$ for Normal distribution

c) Let $$\displaystyle {{x}_{0}}=\mu =0$$ and $$\displaystyle {{\gamma }^{C}}=1$$ where $$\displaystyle {{\gamma }^{C}}$$ is the half width of $$\displaystyle C({{x}_{0}},\gamma )$$. Find $$\displaystyle {{\sigma }^{1}}$$ such that the half width of normal distribution $$\displaystyle {{\gamma }^{G}}=1$$. Then plot $$\displaystyle C(0,1)$$ and $$\displaystyle N(0,{{\sigma }^{1}})$$

d) Find $$\displaystyle \left\{ Q_{1}^{C},Q_{3}^{C} \right\}$$ for $$\displaystyle C(0,1)$$ and $$\displaystyle \left\{ Q_{1}^{G},Q_{3}^{G} \right\}$$ for $$\displaystyle N(0,{{\sigma }^{1}})$$. Then plot them with comments on results.

a) Three quartiles for Cauchy distribution
From the WolframAlpha we have got the general expression for the quartiles $$\displaystyle Q_{1}^{C}$$ and $$\displaystyle Q_{3}^{C}$$:

We can prove this by finding the cumulative distribution function of Cauchy distribution,

Then,

Similarly,

b) Three quartiles for Normal distribution
From the WolframAlpha we can find the quartiles for the normal distribution,

And also we can find the cumulative of normal distribution,

Then we can check the validity of the quartiles given.

c) Compare Cauchy and Gauss distribution with same half width.
With the information given we have our Cauchy distribution with half width equals 1,

To find a normal distribution we need to know the maximum value of

Hence,

Then we must find the x coordinates in the half width nodes.

The difference between the 2 roots should be twice the half width which is 2, hence,

which means the normal distribution is,

The plots of them can be achieved by WolframAlpha



d) Find quartiles for particular Cauchy and Normal distribution. Then plot them with comments on results.
By the formula from part a) and part b) we have,

The following figure is plotted by Adam Franklin,



Corrsponding Matlab Code

Comments

From the results above we can see that the range from $$\displaystyle Q_{1}^{C}$$ to $$\displaystyle Q_{3}^{C}$$ is significantly larger than that from $$\displaystyle Q_{1}^{G}$$ to $$\displaystyle Q_{3}^{G}$$. The probability outside (-0.2865, 0.2865) decrease drastically to very small in normal distribution while in Cauchy distribution this is until (-1, 1). This means the Cauchy distribution allow more probability on the large outcome to occur.