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Math 20: Probability

About

probability distributions

python graph gallery

reproducible python code

main data analysis and math tools:

  • Numpy
  • Pandas
  • Math
  • Scipy

main ploting tools:

  • Matplotlib
  • Seaborn
  • Plotly

Original code and data are in the Github Repository. Both the web page and the code will be updated irregularly.

Last updated: July 10

Slides 0629

  • How Much Does a Hershey Kiss Weight?

figure_hershey_dist(n, fsize, fs)

Slides 0701

  • US States

us_map_html(dict_state, target, title = None, country = 'US', cmap = tealrose)
us_map_png(dict_state, target, title = None, country = 'US', cmap = tealrose)
<iframe src="https://fudab.github.io/math20/figures/US_map_population.html" width="450px" height="300px" scrolling="no" frameBorder="0"></iframe> <iframe src="https://fudab.github.io/math20/figures/US_map_area.html" width="450px" height="300px" scrolling="no" frameBorder="0"></iframe>

  • Histograms of Discrete Probability Distributions

  • fundamental distributions

figure_discrete_hist(job = 'coin', fsize = (8, 6), fs = 20)
figure_discrete_hist(job = 'dice', fsize = (8, 6), fs = 20)
figure_discrete_hist(job = 'bino', fsize = (8, 6), fs = 20)
figure_discrete_hist(job = 'poisson', fsize = (8, 6), fs = 20)

  • refinement of $d\omega$ 

figure_discrete_hist(job = 'bino_1', fsize = (8, 6), fs = 20)
figure_discrete_hist(job = 'bino_2', fsize = (8, 6), fs = 20)
figure_discrete_hist(job = 'bino_5', fsize = (8, 6), fs = 20)
figure_discrete_hist(job = 'bino_10', fsize = (8, 6), fs = 20)

  • Riemann Sum of a Function

figure_riemann_sum(job = 'power', fsize = (8, 6), fs = 20)
figure_riemann_sum(job = 'sin', fsize = (8, 6), fs = 20)

  • Exponential Distribution: Density Function and Cumulative Distribution Function

figure_exp_dist(job = 'pdf', fsize = (8, 6), fs = 18)
figure_exp_dist(job = 'cdf', fsize = (8, 6), fs = 18)

Slides 0706

  • Factorial and Stirling's Formula

stirling(n = 10)
figure_stirling(n = 8, fsize = (8, 6), fs = 18)

  • Hat Check Problem (Fixed Point)

hat_check(n = 10)
figure_hat_check(n = 15, fsize = (12, 6), fs = 20)

Slides 0708

  • Pascal's Triangle

from scipy.special import comb
pascal(n = 10, j = 5)

Quiz 4

  • Fixed Points

fixed_points(n = 6, printing = True)
figure_fixed_points(n = 6, fsize = (8, 6), fs = 18)

Slides 0710

  • One Dimensional Random Walk

random_walk_1D(n = 10, p = 0.6)
path_rw_2D(n = 10, p = 0.6, fsize = (8, 6), fs = 18, index = 1)

  • Two Dimensional Random Walk: A Homework Problem

  • Stock Market

import yfinance as yf
load_STOCK_raw(company_index = 'AAPL')
figure_stock_price(company_index = 'AMZN', date_initial = datetime.date(int(2020),int(6),int(1)), 
                   fsize = (12, 6), fs = 20)

  • Two Dimensional Random Walk (Take-Home Problem)

random_walk_2D(n = 500, p_x = 0.5, p_y = 0.5)
path_rw_2D(n = 500, p_x = 0.5, p_y = 0.5, fsize = (8, 8), fs = 18, index = 1)

Slides 0717

  • Two Dimensional Normal Distribution: Density Function and Contour curve

figure_normal_2d(ind = True, fsize = (10, 6), fs = 20)
figure_contour_normal_2d(ind = True, fsize = (10, 6), fs = 20)

Midterm 1

  • Random Walk Model for Stock Price

random_walk_sp(n = 30, p = 0.6, c = 100, u = 1.1, d = 0.9)
path_rw_sp(company_index = 'DIS', date_initial = datetime.date(int(2020),int(6),int(1)), p = 0.6, u = 1.1, d = 0.9, 
           fsize = (12, 6), fs = 20, index = 1)

Slides 0727

  • Binomial Distribution Versus Poisson Distribution

binomial_poisson_figure(n = 10, p = 0.2, fsize = (8, 6), fs = 18, tag = 1)

  • Sample With or Without Replacement

scatters(n = 3, k = 5, fsize = (10, 10), fs = 20)

Slides 0729

  • Continuous Uniform Distribution

figure_continuous_uniform(fsize = (10, 6), fs = 20)

  • Exponential Distribution

figure_exponential(fsize = (10, 6), fs = 20)

  • Normal Distribution

figure_normal(fsize = (10, 6), fs = 20)

  • 3-$\sigma$ Principle

figure_normal_sd(mu = 0, sigma = 1, c = 1, fsize = (10, 6), fs = 20)

  • Binomial Distribution Versus Normal Distribution

binomial_normal_figure(n = 10, p = 0.2, fsize = (10, 8), fs = 20, tag = 1)

Slides 0803

  • How to Estimate the Quality of Hershey Kisses?

figure_hershey_var_dist(sigma = 0.001, fsize = (8, 6), fs = 18)

Slides 0807

  • Cauchy Distribution

figure_cauchy(fsize = (10, 6), fs = 20)

  • Pokemon

figure_pokemon(df = data_pokemon, column_A = 'Attack', column_B = 'Defense', job = 'scatter', fs = 16)
figure_pokemon(df = data_pokemon, column_A = 'Attack', column_B = 'Defense', job = 'hex', fs = 16)
figure_pokemon(df = data_pokemon, column_A = 'Attack', column_B = 'Defense', job = 'kde', fs = 16)

  • Red Wine

fig = figure_wine(df = data_wine, column_A = 'fixed acidity', column_B = 'density', job = 'reg', fs = 16)

  • Question 5 in Quiz 10

question_5_quiz_10(fsize = (10, 8), fs = 18)

Slides 0817

  • Roll a dice $n$ times.

dices(n = 10)
figure_dices(n = 5, fsize = (8, 6), fs = 18)

  • Sum of Independent Random Variables.

convolution(dist_1 = 'uniform', dist_2 = 'normal', interval = [-3, 3])
figure_convolution(dist_1 = 'uniform', dist_2 = 'uniform', interval = [-0.2, 1.1], fsize = (10, 6), fs = 20)

Slides 0819

  • Law of Large Numbers for a Discrete Random Variable

LLN_experiment(experiment = "coin", n = 10, p = 0.5)
LLN(experiment = 'coin', n = 10, p = 0.5)
figure_LLN(experiment = 'coin', n = 10, p = 0.5, fsize = (8, 6), fs = 18)

Coin Tossing

Dice Rolling

A General Bernoulli Trials Process

  • Law of Large Numbers for a Continuous Random Variable

LLN_experiment_continuous(experiment = "normal")
LLN_continuous(experiment = "normal", n = 10)
figure_LLN_continuous(experiment = 'normal', n = 10, fsize = (8, 5), fs = 18)

Unifrom Density

Normal Density

  • Monte Carlo Method

monte_carlo(n = 100, fsize = (6, 6), fs = 17)

Slides 0821

  • Central Limit Theorem for Bernoulli Trials

figure_bino_hist(job = 'original', p = 0.5, fsize = (12, 8), fs = 20)
figure_bino_hist(job = 'standardized', p = 0.5, fsize = (12, 8), fs = 20)
figure_bino_hist(job = 'standardized_normal', p = 0.5, fsize = (12, 8), fs = 20)

  • $k$ and $b(n, p, k)$

  • $x$ and $b(n, p, k)$

  • $x$ and $\sqrt{npq}b(n, p, k)$

Progress

The dates are consistent with the class.

  • 0629
  • 0701
  • 0706
  • 0708
  • quiz 4
  • 0710
  • 0717
  • Midterm 1
  • 0727
  • 0729
  • 0803
  • 0807
  • 0817
  • 0819
  • 0821