# Probability of event using Central Limit Theorem + plotting results

I've been doing a Udemy course called: "Statistics for Data Science" and I decided to solve one of the homework with Python to kill two birds with one rocket #elon.

The team of traders under your supervision earns profits which can be approximated with Laplace distribution. Profits (of any trade) have a mean of $95.70 and a std. dev. of$1,247. Your team makes about 100 trades every week.

Questions:

A. What is the probability of my team making a loss in any given week? B. What is the probability of my team making over $20,000 in any given week? As I just started to learn Python I would be happy for some hints and opinions. # set up import math import scipy.stats as st import matplotlib.pyplot as plt import numpy as np # Data mu = 95.7 # mean sigma = 1247 # standard deviation n = 100 # sampling size (trades here) xcritical1 = 0 # making a loss xcritical2 = 20000 / n # Earning$20k a weak by 100 trades

mu_1 = mu                           # Based on Central Limit Theorem
sigma_1 = sigma / (math.sqrt(n))    # Based on CLT

# Calc
def Z(xcritical, mu, sigma):
return (xcritical - mu) / sigma     # Standard Score (z-value)

Z1 = Z(xcritical1, mu_1, sigma_1)
Z2 = Z(xcritical2, mu_1, sigma_1)

P1 = st.norm.cdf(Z1)                    # Cumulative Distribution Function for ND
P2 = 1 - st.norm.cdf(Z2)

print('A. Probability of making loss in any given week is', '{0:.4g}'.format(P1*100) + '%')
print('B. Probability of making over $20k in any given week is', '{0:.4g}'.format(P2*100) + '%') # Plots def draw_z_score(x, cond, mu, sigma, title): y = st.norm.pdf(x, mu, sigma) # Probability Density function for ND z = x[cond] plt.plot(x, y) plt.fill_between(z, 0, st.norm.pdf(z, mu, sigma)) plt.title(title) plt.text(-300, 0.0020, r'$\mu=' + str(mu_1) + ',\ \sigma=' + str(sigma_1) + '$') plt.show() x = np.arange(-400, 500, 1) # Fixed interval by experimenting title1 = 'Probability of making loss: ' + '{0:.4g}'.format(P1*100) + '%' title2 = 'Probability of earning more than$20k: ' + '{0:.4g}'.format(P2*100) + '%'

draw_z_score(x, x < xcritical1, mu_1, sigma_1, title1)
draw_z_score(x, x > xcritical2, mu_1, sigma_1, title2)

• Why are you using a normal distribution, instead of a Laplace distribution as stated in the task description? – Graipher Feb 12 '18 at 6:13
• Because any distribution can be approximated by Normal Distribution by Central Limit Theorem and this is what they expected me in this task. – Mateusz Konopelski Feb 12 '18 at 14:48
• Fair enough (if you mean "the sum of random variables from any random distribution", instead of "any random distribution"). And in this case the approximation is actually already good enough with N=100: repl.it/@graipher/Sumlaplace-vs-Gauss – Graipher Feb 12 '18 at 15:30

You seem to know about str.format (since you use it), yet you are still doing string addition:

title1 = 'Probability of making loss: ' + '{0:.4g}'.format(P1*100) + '%'
title2 = 'Probability of earning more than $20k: ' + '{0:.4g}'.format(P2*100) + '%'  Just make it a single string: title1 = 'Probability of making loss: {0:.4g}%'.format(P1*100) title2 = 'Probability of earning more than$20k: {0:.4g}%'.format(P2*100)


You could also use the % formatting option for floats (described in the fifth table here), which automatically multiplies with 100 and adds the % sign. The number of digits after the decimal point work differently, though, compared to the g format:

"{:.4%}".format(0.5)
# '50.0000%'
"{:.4g}%".format(0.5*100.)
# '50%'
"{:.4g}%".format(0.5123123*100.)
# '51.23%'
"{:.4%}".format(0.5123123)
# '51.2312%'


You should also re-organize your code according to the following scheme:

import something

GLOBAL_CONSTANT = "value"

class Definitions

def functions()

if __name__ == '__main__':
main()
# or some small code using the stuff defined above


That last part is there to protect your code from being executed if you want to import some part of this script from another script. At the moment if you want to do from laplace import Z, your whole code would run. With the if __name__ == '__main__' guard, it will only run when you execute this script.

You should add docstrings to your functions, describing what they do, what arguments they take and what they return. Have a look at PEP257 for the guidelines regarding docstrings.

Yes, it is justified to use the normal distribution here, since the Central Limit Theorem guarantees that the sum of random variables tends towards a normal distribution. The approximation is quite good for $N = 100$, as can be seen with the following small script:

import numpy as np
import matplotlib.pyplot as plt

mu = 95.7
sigma = 1247.
n = 100    # how many random variables to sum for each value
N = 1000    # how many values to generate

x_l = np.random.laplace(mu, sigma/np.sqrt(2), (n, N)).sum(axis=0)
x_g = np.random.normal(n*mu, np.sqrt(n)*sigma, N)

plt.hist(x_g, label="gaus")
plt.hist(x_l, label="sum(laplace)", histtype='step')
plt.legend()
plt.show()