Mastering Probability & Statistic Python (Theory & Projects)

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.

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

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.

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.

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.

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.

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.

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

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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].

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.

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.