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)

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)

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

</table>

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.

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