Mastering Probability & Statistic Python (Theory & Projects)

Compare probability and statistics: probability looks forward to predict future likelihood using laws, while statistics analyzes past data to derive rules guiding future predictions.

Define a set as an unordered collection of distinct, well-defined objects, and introduce its core terminology and properties, including elements and the notions of finite, countable, and uncountable sets.

Define the elements or members of a set and the membership notation. Describe finite and infinite sets, the empty set, and cardinality, including countable versus uncountable examples.

Explore subsets of a parent set, including the empty set. Define the power set as all subsets and identify the universal set as the container for discussed objects.

Define sets in python, practice membership tests with in, and check subsets with issubset or custom logic, then generate and print the powerset.

Generate a power set in Python using NumPy arrays and binary boolean indices to produce all subsets and understand the 2^n cardinality.

Explore core set operations: union, intersection, and difference with examples. Learn about complements, universal set, De Morgan's laws, partition, disjoint sets, and Venn diagrams.

Explore the union and intersection of a set with an empty set, examining what happens when you unite a with the empty set and what the intersection with the empty set yields.

Explore set operations with the empty set, showing union returns the original set and intersection yields an empty set, while avoiding duplicates.

Explore the operation difference between a set and an empty set using example A = {1, 9, 7, 13}, including A minus {5} and related exercises.

Explain the set difference by selecting elements from the first set not present in the second set, with cases where the second set is empty.

Count the number of ways to partition a 10-element set into two nonempty disjoint subsets whose union is the whole set, illustrating B and C with examples.

Count two-set partitions of a 10-element set into sizes 1–9, 2–8, 3–7, 4–6, and 5–5 using combinations and basic counting methods in probability theory.

Practice Python set operations by building universal omega, defining A and B, and performing union, intersection, difference, and complements with numpy. Verify de Morgan's law.

Explore how Venn diagrams visualize set operations like union, intersection, difference, and complement, and see how subsets, partitions, and the universal set illustrate probability foundations.

Explore the properties of sets and ordered collections, implement a Python is_partition function to test partitions and set complements, and verify identities through randomized experiments.

Learn how an experiment produces one of several outcomes and why it is random, with examples like tossing a fair coin and rolling a four-sided die.

Identify outcomes as results of an experiment and define the sample space as all possible outcomes. Illustrate with coin tosses, two-coin experiments, and finite or infinite, countable sample spaces.

Analyze the sample space of an experiment with three rolls of a four-sided die and one coin toss, identifying all possible outcomes and their structure.

Enumerate the sample space for an experiment with three rolls of a four-sided die and one coin toss, and show it contains 128 elements.

Define an event as a subset of the sample space, including the empty set and full space. Use two dice and even-sum outcomes as examples.

For a sample space of size 16, count all subsets as events, including the empty set. There are 65536 possible events.

Explore how the powerset of a set forms possible events, with each subset as an event, and note that a 16-element sample space yields 2^16 events, including the empty set.

This exercise reinforces treating events as sets and identifying disjoint events, with hints about renaming the event set to clarify probability reasoning.

Learn how disjoint events form disjoint sets, where the intersection is empty, illustrated by events like one-three and two-four.

Explore how to design a sample space and assign likelihoods to events, building a probability model that supports unambiguous predictions.

Define probability with three axioms: non-negativity, additivity for disjoint events, and the sample space equals one. These imply probabilities lie between zero and one, with zero allowed.

Derive probability axioms from basic concepts like the sample space, events, and complements. Apply union, intersection, and complement rules to show P(A^c)=1−P(A) and P(A∪B)=P(A)+P(B)−P(A∩B), with A⊆B implying P(A)≤P(B).

Explore whether an empty seat can be a non-zero probability event by focusing on the probability axioms.

Demonstrates that the empty event has zero probability. Uses the axiom that the sample space has probability one and the union of disjoint events equals the sum of their probabilities.

Study a discrete probability model with two four-sided dice, using a 16-outcome table to determine events like even sums and at least one die showing four under a probability law.

Compute the probability that a next patient has neither malaria nor typhoid by using complements, union and intersection with De Morgan's law and the probability axiom.

Explore the difference between discrete and continuous probability models, noting that discrete models have countable sample spaces with single-outcome probabilities, while continuous models are uncountable and assign probabilities to intervals.

Explore conditional probability by showing how partial information, such as a sum of nine, a negative medical test, or radar readings, changes the likelihood of outcomes.

Delve into conditional probability using a loaded six-die example to define events A and B, compute probabilities, and derive P(A|B) with a concrete probability law.

Derive and apply the conditional probability formula P(A|B) = P(A∩B)/P(B) using dice outcomes, normalized by B, and recognize A independent of B when P(A|B) = P(A).

Explore how conditional probability underpins almost all machine learning models, using random variables, probability distribution, and real-world examples like face recognition, video activity recognition, and text to speech.

Learn law of total probability, using a partition of the sample space to sum probabilities of B across disjoint A_i, and derive marginal distributions from joint distributions of random variables.

Explore statistical independence in probability, showing that A and B are independent when P(A∩B)=P(A)P(B), and discuss conditional probability A|B, symmetry, and independence of all subsets.

Examine independence and conditional independence, proving that if A does not depend on B, then B does not depend on A, and introduce Bayes' rule and Bayesian classifiers.

Provide examples of two events that are dependent, but become independent when conditioned on a third event, illustrated with A, B, and C.

Examine conditional independence in a probability model using a coin-picking experiment with a fair and a two-headed coin; show A and B are dependent overall but independent given C.

Explore Bayes rule and its simple, powerful formula for P(A|B), its role in classification and generative versus discriminative modeling, and key components like class conditional distribution and prior.

Investigate how real data are represented as random variables and how Bayes' rule, prior distributions, and a naive Bayes classifier enable predictions using class-conditional distributions.

Build a probability model for a simultaneous four-sided die roll and two coin flips, then compute the probability that the die shows an even number and both coins land heads.

Explore how a random variable is a real-valued function of outcomes, illustrated by rolling two dice to map outcomes to sums, maxima, and a prime-number indicator, with probability laws.

Define random variables from a two-dice experiment, including the sum and the maximum; map values to event probabilities and use a binary prime-sum variable to introduce the probability mass function.

Examine whether zero probability for a random variable value indicates an empty event, and distinguish between empty events, impossible events, and events with probability zero.

Explore how discrete and continuous random variables treat the event X equals A, noting discrete yield an empty event and continuous yield zero probability for non-empty sets.

Define discrete random variables and the probability mass function, introduce Bernoulli variables (0 or 1) via coin toss examples, and preview Python-based simulations for fair and biased coins.

Mastering probability and statistics with python, this lecture builds a bernoulli trial simulator using numpy to estimate the coin's probability of success by counting heads and tails over many trials.

Determine if the next ball outcome, a batsman hitting a six or not, can be modeled as a Bernoulli random variable in this Python practice exercise.

Model a simple experiment with Bernoulli random variables, using X for a win and Y for a six, each with its own probability, to represent binary outcomes.

Explore independent bernoulli trials, build pmf for a geometric random variable, and model the number of coin tosses until the first head using a biased coin.

Explore the geometric random variable from independent Bernoulli trials, where X is the number of coin tosses until the first head, and show the probabilities sum to one.

Define a geometric trial in Python that returns the total number of Bernoulli trials until the first success, and explore its distribution using data, histograms, and varying probability of success.

Explore the binomial random variable arising from independent bernoulli trials, counting heads in n coin tosses. Derive its pmf using combinations, p, and 1-p, and preview Python simulations.

Learn to implement a binomial trial in Python, generate Bernoulli trials, and visualize how probability of success and trial count shape the binomial distribution and its mean.

Explore how real data sets map to random variables, including discrete and continuous types, and model joint and conditional distributions for machine learning classification and regression tasks.

Extend your understanding of discrete random variables beyond binomial and geometric by identifying another famous discrete variable in real datasets through an interactive exercise.

Explore the Poisson distribution, its PMF with lambda > 0, and how it models discrete random variables; relate it to binomial cases and Gaussian approximations in real datasets.

Work through homework problems: model the maximum of three rolls of a four-sided die with a uniform pmf, and compute the probability that the maximum is even.

Explore continuous, uncountable random variables on the interval [0,1], why individual values have zero probability, and how probabilities live on intervals via density functions.

Investigate whether the random variable x, defined as the midpoint of one of six disjoint subintervals of a line segment chosen by rolling a six-sided die, is continuous.

Determine that X, the midpoint of a randomly selected disjoint interval from six intervals, is a discrete random variable with a finite set of midpoints, not a continuum.

Learn how probability density functions model continuous variables by turning interval areas under the curve into probabilities, with total area one and densities that may exceed one.

Explore the properties of a valid probability density function for a continuous random variable X, focusing on the criteria and constraints that define its validity in this exercise.

Define a valid probability density function by ensuring nonnegativity and normalization, where the area under the curve equals one; the domain can be any real values for X, including negatives.

Explore the uniform distribution on the interval 10 to 30, with density 1/20 inside and zero outside, and compute probabilities via area under the curve in Python using NumPy.

Explore whether the random variable X, representing a fair die roll with equally likely outcomes, forms a uniform distribution, illustrated through a uniform random variable exercise.

Identify that the dice roll X is a discrete uniform random variable with equal PMF entries of 1/6 for outcomes 1 through 6, demonstrating that uniform distributions can be discrete.

Generate thousand uniform random numbers with NumPy, observe a flat distribution between zero and one, scale to zero–hundred or shift to twenty–120, and visualize with a histogram and kde.

Explore the exponential distribution, a continuous nonnegative variable with density lambda e^{-lambda x} for x >= 0, and see how lambda shapes the distribution and its normalization.

Explore how the lambda parameter shapes the exponential distribution's density, and how changing lambda affects the arrival rate and distribution behavior, with examples using 0.5, 0.7, and 5.9.

Explore how varying lambda values shape exponential distributions: higher lambda yields faster decay and taller peaks, while smaller lambda slows decay with subtler peaks.

Explore the exponential distribution by generating x values and plotting the density f(x) for varying lambda, observing how higher lambda sharpens decay while the area under the curve remains one.

Explore the gaussian (normal) distribution as a continuous random variable with mean mu and standard deviation sigma. Learn its gaussian pdf, normalization, and how mu and sigma shape the density.

Analyze how sigma affects a Gaussian distribution, comparing outcomes for large versus small sigma in this exercise.

Examine how sigma, the variance, shapes gaussian random variables: larger sigma flattens the bell curve, while smaller variance sharpens the peak, with a note on the exponential distribution's parameter effects.

Generate and visualize Gaussian distributions by adjusting mu and sigma. Use histograms, density plots, and kernel density estimates to compare and discuss parametric versus nonparametric estimation on data like iris.

Explore how transforming discrete and continuous random variables into new features enables dimensionality reduction and stronger probability models for classification and regression tasks.

Explore the cumulative distribution function (cdf), why it is useful, and how it relates to both discrete and continuous variables, with a homework task to research online.

Learn how to compute the expectation, or mean, of a random variable using discrete summation and continuous integration, with Bernoulli examples and a Python demo.

Compute the sample mean across Bernoulli, geometric, and binomial experiments to see how it approximates the distributions' expected values and reveals parameter meanings under the law of large numbers.

Understand the law of large numbers: as the sample size grows for iid data, the sample mean converges to the distribution's expected value.

Explore the law of large numbers by simulating iid data from Bernoulli, geometric, binomial, and normal distributions, and observe how the sample mean converges to the true mean.

Explore how transforming a random variable affects its expected value using the probability mass or density function. Understand moments and variance, including second moment and normal distribution mu and sigma.

Solve the homework on the maximum of three rolls of a four-sided die with a uniform pmf, computing the expected value, variance, and the fourth moment.

Build a Bayes classifier from scratch for iris data using a single feature, estimate priors and gaussian class-conditional parameters, and evaluate with train-test splits.

Learn to build joint distributions for multiple random variables, including joint pmf and joint density, derive marginal distributions via the law of total probability, and explore the multivariate Gaussian distribution.

Explore deriving the expectation for two discrete random variables X and Y using their joint BMF, then extend to continuous distributions by replacing sums with integrals.

Compute the expected value of Z = X + Y via joint PMF and linearity of expectation, and derive X and Y marginals to show E[Z] = E[X] + E[Y].

Compute the expected value of a binomial random variable X with parameters n and p, where n independent Bernoulli trials have success probability p, giving E[X] = n p.

Discover how to derive the expected value of a binomial variable by expressing it as a sum of independent Bernoulli trials and summing their expectations, yielding E[X] = n p.

Derive the expectation of the product of two independent discrete random variables by using their joint distribution; compute E[XY] for X and Y with a joint pmf.

Master independent X and Y: compute E[XY] using the joint distribution as the product of the individual distributions, and show E[XY] = E[X]E[Y] for discrete variables.

Explore the multivariate Gaussian distribution for a random vector, its density with the mean vector and covariance matrix, and its role in data science and machine learning.

Learn conditioning of random variables across discrete and continuous cases, using P(X|Y)=P(X,Y)/P(Y) and joint over marginal distributions. Understand independence and conditional independence to prepare for classification and regression topics.

Build a probability model for classification by predicting the discrete class variable y from the feature vector x. Explore generative vs discriminative modeling, p(x|y) vs p(y|x), with Gaussian options.

Explore how the naive Bayes classifier uses the conditional independence assumption to simplify estimating the joint density of features given the class, and apply Bayes theorem to predict the class.

Learn regression for a continuous target by using conditional density and the expected value of y given x, with linear regression and the curse of dimensionality in focus.

The curse of dimensionality shows that estimating a joint distribution requires vast data as variables increase, making high-dimensional bins unreliable; PCA is mentioned as a remedy.

Implement a from-scratch naive bayes classifier in Python using the iris data from Seaborn, assuming independence of features, perform a train-test split, and report the joint distribution and classification results.