# Eleisha's Segment 11: Random Deviates

To Calculate:

1. For the Cauchy distribution (Segment 8, Slide 3), find the inverse function of the CDF.

You can easily use mathematica to find the inverse function of the CDF for the Cauchy distribution.

OR You could just you pencil and paper...

$\displaystyle P(x) = \frac{1}{2} + \frac{1}{\pi}\arctan\left[ \frac{x - \mu}{\sigma} \right]$

Let $\displaystyle P(x) = P$

Now we just find the inverse of the function:

$\displaystyle P - \frac{1}{2} = \frac{1}{\pi}\arctan\left[ \frac{x - \mu}{\sigma} \right]$

$\displaystyle \pi\left( P - \frac{1}{2} \right) = \arctan\left[ \frac{x - \mu}{\sigma} \right]$

$\displaystyle \tan\left[\pi\left( P - \frac{1}{2} \right)\right] = \frac{x - \mu}{\sigma}$

$\displaystyle \sigma \tan\left[\pi\left( P - \frac{1}{2} \right)\right] + \mu = x$

So the CDF, $\displaystyle F(P)$ is:

$\displaystyle F(P) = \sigma \tan\left[\pi\left( P - \frac{1}{2} \right)\right] + \mu$

2. In your favorite programming language, write a function that returns independent Cauchy deviates.

Here is some python code that generates independent Cauchy deviates:

import math
import random

mu = 0
sigma = 1
N = 10
#The inverse of the Cauchy CDF
def get_inv_cauch_cdf(p_y, mu, sigma):
cdf_val = mu + sigma*math.tan((p_y - 0.5)*math.pi)
return cdf_val

#This is a function that will generate an independent Cauchy deviate using the Transformation
#Method
def get_variate():
x = random.random()
y = get_inv_cauch_cdf(x, mu, sigma)
return y

def get_cauchy_deviates(N):
deviates = []
for i in xrange(0, N):
deviates.append(get_variate())
return deviates

rand_deviates = get_cauchy_deviates(N)

print str(N) + " Random Deviates from a Cauchy Distribution: "
print rand_deviates


Here is some same output generated by the code above:

10 Random Deviates from a Cauchy Distribution:
[9.0851283407409813, -0.65383372885854252, -6.7627447406781362, -1.8271582934958259, 0.57399828693079358, -0.54025381399970152, -2.5448556887215492, 0.96565869875860988, 0.80155776954380109, 0.56424562260925804]


1. Suppose you want a function that returns deviates for Student($\displaystyle \nu$ ). Could you use the Cauchy pdf (or some scaling of it) as a bounding function in a rejection method? How efficient is this (i.e., what fraction of the time does it reject)? 2. Explain the three inequality tests in the "while" statement in Leva's algorithm (slide 7) and why they are hooked together with logical operators in the way shown.

Class Activity

Group: Eleisha Jackson and Daniel Shepard

During class, we attempted to develop our own (pseudo) random number generator. Our method: In order to create a generator that uniformly sampled between the range of zero and one we used a linear algorithm based on the the time as a random seed. Using the time as input, we developed a mathematical algorithm that generated a sequence of random numbers. The first number in the sequence: $\displaystyle x_0$ was a deterministic value that was calculated using the time. Then every $\displaystyle x_k$ afterwards was a generated as a function of $\displaystyle x_{k-1}$ .

Here is our MATLAB code:

function [x] = randUniform(N)
t = clock;
m = 2^48;
a = 25214903917;
c = 11;
sec = t(6);
seed = sec*m/60;
y = size(N,1);
y(1) = mod(a*seed,m);
for(k = 2:N)
y(k) = mod(a*y(k-1) + c,m);
end
x = y/m;



Using this method we were able to verify that out random generator equally sample the range between (0, 1). Unfortunately our random number generator did not pass the required p-value test. We believe this is because each number in the sequence is dependent on the initial starting number $\displaystyle x_0$ and therefore can be predicted. This means that each number in the randomly generated sequence of 1 million points that we test are not independent.

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