Random Systems

problems

(6.1)

(a) Work out the first three cumulants C1, C2, and C3.

(6.2)

(c) Write a uniform random number generator, and transform it by equation (6.78). Numerically evaluate the first three cumulants of its output.

import numpy as np
import time
import matplotlib.pyplot as plt
import math

# Linear Congruential Random Generator

prev = 0

a = 8121
b = 28311
c = 134456
 
def random_seed(x):
    global prev
    prev = (int(x)*a + b) % c

def random_gen():
    global prev
    prev = (a * prev + b) % c
    return prev / c

print(prev)

0

# Simulated LFSR

prev2 = ''
size = 32
taps = [0,1,21,31]

def random_seed2(size):
    global prev2
    curr_time = int(time.time()%1*(10**16) % (2**size))
    prev2 = format(curr_time, 'b').zfill(size)

def random_gen2():
    global prev2
    bit = 0
    for t in taps:
        bit = (bit + eval(prev2[t])) % 2
    prev2 = str(bit)+prev2[:-1]
    return int(prev2,2) / (2**size)

random_seed2(size)
print(prev2)
random_gen2()
print(prev2)
random_gen2()
print(prev2)

10001110101110010110011010100000
01000111010111001011001101010000
10100011101011100101100110101000

# 

N = 100

print(prev)
random_seed(round(time.time()%1*(10**6)))
print(prev)

x1 = np.array([random_gen2()*2*math.pi for i in range(N)])
x2 = np.array([random_gen2() for i in range(N)])

0
23500

plt.scatter(x1, x2)
plt.xlabel("x1")
plt.ylabel("x2")
plt.show()

concepts

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1. properties of a random variable

• near-certain probabiliy ( $1-\epsilon$ ) vs near-random uncertainty ( $ 0.5 + \epsilon $ )
• it is not possible to predict the value of a random variable, but it is possible to predict the probability for seeing a given event in terms of the random variable.
  • realization: each instantiation of identical stochastic procedure will obey the same distribution laws.
• discrete vs. continuous;
• probability(|x-target value| < tolerance)
  $ \int_{a}^{b} p(x) dx $
• support: values over which the integral of probability function is defined
• expectation value: \begin{equation*} \begin{aligned} \langle f(x) \rangle & = \lim_{N\to\infty} \frac {1}{N} \sum_{i=1}^{N} f(x_i) \\ & = \int_{-\infty}^{\infty} f(x)\,p(x)\;dx \end{aligned} \end{equation*}
  • mean value: $ \bar{x} \equiv \langle x \rangle $
• maximum likelihood value?
• variance $ \sigma^2_x \equiv \langle (x-\bar{x})^2 \rangle = \langle x^2 \rangle - \langle x \rangle^2$: average value of the square
• *standard deviation

2. inter-relationships among a collection of random variables

\begin{equation} p(x|y) = \frac {p(x,y)}{p(y)} \end{equation}
Bayes's rule: the joint probability of x,y is equal to the probability of x given y multiplied by probability of y

• independent : $ \begin{aligned} p(x|y) &= p(x) \\ p(x,y) &= p(x)p(y) \end{aligned} $

3. stochastic processes: random systems that evolve in time

4. algorithms for generating random numbers

facts

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procedures

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