Introduction to Probability for Data Science Book Club
Welcome
Book club meetings
Pace
Resources
1
Mathematical Background
Infinite Series: Warmup 1
Infinite Series: Warmup 2
Geometric Series
Binomial Series
Binomial Series: Code
Taylor Approximation
Taylor Approximation: Example
Exponential Series
Integration
Fundamental Theorem of Calculus
Linear Algebra
Linear Algebra Example
Weighted Norms
Matrix Calculus
Combinatronics
Meeting Videos
Cohort 1
2
Probability
Motivating example
2.1
Set Theory
\(\Omega\)
and
\(\emptyset\)
Set Operations
Important Properties
Disjoint and Partition
2.2
Probability Space
Sample Space
\(\Omega\)
Event space
\(\mathcal{F}\)
Probability law
\(\mathbb{P}\)
2.3
Axioms of Probability
Corollaries
2.4
Conditional Probability
Example
Independance
Bayes’ Theorem and law of total probability
Example: Monty Hall Problem
Monty Hall Problem: Simulation
Monty Hall Problem: Alternative assumptions
Meeting Videos
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3
Discrete Random Variables
3.1
Random Variables
Probability measure on random variables
3.2
Probability Mass Function
PMF and probability measure
Normalization property
PMF vs histogram
3.3
Cummulative Distribution Functions (Discrete)
Properties of the CDF
Converting between PMF and CDF
3.4
Expectation
Existence of expectation
Properties of expectation
Moments and variance
3.5
Common Discrete Random Variables
Bernoulli Random Variable
Properties of Bernoulli random variables
Binomial random variable
Geometric random variable
Poisson random variable
Meeting Videos
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4
Continuous Random Variables
4.1
Probability Density Function
More in-depth discussion about PDFs
Connecting with the PMF
4.2
Expectation, Moment and Variance
Definition and properties
Moment and variance
Existence of expectation
4.3
Cumulative Distribution Function
CDF for continuous random variables
Properties of CDF
Retrieving PDF from CDF
CDF: Unifying discrete and random variables
4.4
Median, Mode and Mean
Median
Mode
Mean
4.5
Uniform and Exponential Random Variables
Uniform random variables
Exponential random variables
4.6
Gaussian Random Variables
Standard Gaussian
Skewness and kurtosis
Origin of Gaussian random variables
4.7
Functions of Random Variables
General principle
4.8
Generating Random Numbers
General principle
Meeting Videos
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5
Joint Distributions
Joint Distributions
Many definitions
Joint CDF
Joint Expectation
Correlation and independance
Conditional PMF and PDF
Example (Practice Exercise 5.8)
Conditional Expectation
Example 5.22
Sum of two random variables
Random vectors
Multidimensional gaussians
Example in R
Transformations of Gaussians
Eigendecomposition
Generating random variables
PCA
PCA example
Meeting Videos
Cohort 1
6
Sample Statistics
SLIDE 1
Meeting Videos
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7
Regression
SLIDE 1
Meeting Videos
Cohort 1
8
Estimation
SLIDE 1
Meeting Videos
Cohort 1
9
Confidence and Hypothesis
SLIDE 1
Meeting Videos
Cohort 1
10
Random Processes
SLIDE 1
Meeting Videos
Cohort 1
Published with bookdown
Introduction to Probability for Data Science Book Club
Chapter 3
Discrete Random Variables
Learning objectives:
Convert events/statements to numbers, via
random variables
.
Assign probabilities to a random variable, via
probability mass function
.
Formalize the
expectation
from an experiment.