Python Mastery for Data, Statistics & Statistical Modeling

Explore probability and statistics with Python through beginner-friendly theory and live coding in NumPy. Connect random variables, sets, and regression to core data science methods, including neural networks.

Meet Kashif Murtaza, an artificial intelligence expert with 15+ years teaching and Python experience, delivering hands-on practice and publications for this Python mastery course in data, statistics, and modeling.

Master the fundamentals of probability and statistics with Python, linking core concepts to machine learning. Explore sets, random experiments, and random variables, with Bayes classifier code on real data.

Explore the difference between probability and statistics, how they build rules from data, and how statistics looks backward to describe past data while probability looks forward to predictions.

Define a set as an unordered collection of distinct, well-defined objects. Differentiate finite, infinite, countable, and uncountable sets, and avoid duplicates in a basic set.

Define sets and their elements, use names for sets, and grasp membership, empty sets, finite and infinite cardinality, countable vs uncountable, and topics like subsets, power sets, and universal sets.

Define subsets with examples, noting the empty set and a set as subsets of themselves, introduce the power set of all subsets, and describe universal set as the discussion universe.

Practice Python sets by defining A and B, checking membership, testing subset relations, and implementing a custom subset function, then explore the power set as a fun exercise.

Generate the power set of a given set in Python using a binary indexing approach with boolean arrays and NumPy, covering all subsets from empty to the full set.

Explore union, intersection, difference and complement operations on sets, learn De Morgan's laws, partitions and disjointness, and visualize with Venn diagrams, plus Python demonstrations.

Explore union and intersection of a set with an empty set, using phi to denote the empty set. See how these operations interact with an arbitrary set A.

Explore why A union empty set equals A, with no duplicates, and why A intersect empty set equals the empty set.

Explore set difference with an empty set and practice computing A minus five and five minus A to understand how emptiness affects results.

Demonstrates the set difference operation by showing that elements of the first set not in the second are copied to the result, as A minus phi equals A.

Explore counting all ways to partition a ten-element set into two nonempty, disjoint subsets whose union is the whole set, with practical examples.

Explains counting two-set partitions of a ten-element set by grouping k elements with the rest, using combinations and counting methods. Covers one-by-nine, two-by-eight, three-by-seven, four-by-six, and five-by-five partitions.

Practice Python set operations in a Jupyter notebook using NumPy to perform union, intersection, difference, complements, and De Morgan's laws, with universal set concepts.

Explore sets and Venn diagrams to visualize union, intersection, difference, and complements within a universal set, using S and T to illustrate subset and partition concepts.

Master core set concepts, including unordered, distinct elements, and partitions into disjoint subsets whose union forms the parent set. Practice Python code to verify set identities and complements.

Define an experiment as a process that yields one of several outcomes under fixed conditions. Show coin tosses and dice as examples to highlight random outcomes.

Define outcomes as experiment results and build the sample space as the set of all possible outcomes, illustrated by coin tosses, first-head stopping, and infinite but countable possibilities.

Explore the sample space of a combined experiment: three rolls of a four-sided die and one coin toss, and enumerate all possible outcomes.

Explore how to enumerate the sample space for an experiment with a four-sided die rolled three times and a coin tossed, revealing 128 possible outcomes.

Define an event as a subset of the sample space, or an element of the power set, including empty set. Show two dice and temperature rules illustrating a random experiment.

Explore how a 16-element sample space yields 2^16 possible events, since every subset—including the empty subset—qualifies as an event in the experiment.

Explain how a set with n elements has a power set of 2^n subsets, each an event; a 16-element space yields 2^16 events, with the empty set.

Justify disjoint events by treating events as sets and identifying when two sets have no shared elements, using the hint to rename the event with set and define disjoint sets.

Renaming or typecasting an event as a set reveals disjoint events as disjoint sets, since their intersection is empty; for example, event 113 and event 24 are disjoint.

Clarifies core experiment concepts—random outcomes, sample space, sample points, and events—applied to a four-sided die. Explore finite versus infinite sample spaces and an event with even sum of rolls.

Design a probability model by choosing a sample space and assigning non-negative probabilities to events, then use the model to make unambiguous predictions.

Explore the three core probability axioms—non-negativity (zero allowed), additivity for disjoint events, and the sample space has probability one. Then apply them to dice outcomes and even-number events.

Derive the probability axioms: P(Aᶜ) = 1 − P(A) and P(A ∪ B) = P(A) + P(B) − P(A ∩ B), noting P(A) ≤ P(B) when A ⊆ B.

Investigate whether the empty set as an event can have non-zero probability by applying the probability axioms and laws of probability.

Proves that the empty set has zero probability, using the probability axiom that the sample space has probability one and the union of disjoint events sums to one.

Learn discrete probability models with two four-sided dice, define the 16-outcome sample space, and evaluate events like even sums (1/2) and at least one four (7/16).

Use probabilistic modeling to assess the probability the next patient has neither malaria nor typhoid, applying De Morgan's law and probability axioms with the given 0.6, 0.7, 0.4.

Distinguish discrete from continuous probability models by contrasting countable versus uncountable sample spaces, then show why probabilities attach to intervals (not single outcomes) with a circle dart example.

Explore conditional probability, the probability of an event given partial information, illustrated by dice, disease testing, and radar examples, showing how information changes likelihood and informs machine learning tasks.

Formalizes conditional probability notation using a loaded six-sided die, derives P(A) and P(A|B) for A not larger than four with B even, and discusses independence.

Derive the conditional probability formula and apply it to events A and B for a die example. Normalize by P(B) to get P(A ∩ B) and note Bayes classifier relevance.

Apply conditional probability to machine learning using random variables and probability distributions. See how face recognition and activity recognition use data X and identity Y in classification and regression.

Learn the law of total probability, or total probability theorem, and derive marginal distributions from joint distributions of several random variables by partitioning the sample space.

Explore statistical independence of events and how A and B being independent means P(A∩B) = P(A)P(B); question whether independence is symmetric and how independence extends to A,B,C via all subsets.

Examine independence and conditional independence: if A does not depend on B, B does not depend on A. Discover how these ideas underpin Naive Bayes and Bayes rule in Python.

Identify three events A, B, and C where A and B are dependent. Then show they become conditionally independent given C and practice calculating P(A∩B|C) = P(A|C)P(B|C).

Demonstrates how two coin-toss events become conditionally independent given the coin choice, despite being dependent unconditionally, using a fair vs two-headed coin example.

Explore Bayes rule and Bayes theorem, and how Bayes classifiers, including Naive Bayes, and generative and discriminative models, rely on class-conditional, prior, and marginal distributions.

Learn how real data are represented as random variables and how Bayes, prior and class conditional distributions, and naive Bayes connect data to predictions.

Build a probability model for a four-sided die and two-coin toss, assuming all outcomes are equally likely, and compute probability that the roll is even and both coins show heads.

A random variable is a real-valued function of an experiment outcome, illustrated by rolling two dice to map outcomes to sums, maximum, or a prime-sum indicator, with probabilities from events.

Define random variables from rolling a die twice, including the sum and the maximum. Use a binary variable for prime sums and tie values to underlying events and probabilities.

Clarify whether a zero probability for a specific value of a random variable corresponds to an empty event.

Explain how x equals a can be impossible empty event for discrete variables, while for continuous variables, probability that x equals a is zero and may correspond to nonempty sets.

Explore the probability mass function for discrete random variables, with Bernoulli examples and the role of probability laws, sample space, biased versus fair coins, and logistic regression.

Develop a Python simulation of a Bernoulli random variable by implementing a Bernoulli trial with NumPy, and estimate the probability of success through repeated trials.

Explore whether the next ball outcome being a six can be modeled as a Bernoulli random variable, linking a six to team A's win.

Model this experiment with a Bernoulli random variable by defining x as 1 if team a wins and 0 otherwise, and y as 1 if next ball is a six.

Explore independent Bernoulli trials with a biased coin, define the geometric random variable as the number of tosses until the first head, and derive its PMF.

Learn how a geometric random variable counts coin tosses until the first head in independent Bernoulli trials, and verify normalization via p and q with q=1-p.

Explore geometric random variables in Python by simulating Bernoulli trials to count the number of trials until the first success, and visualize the distribution with histograms.

Explore binomial random variables from independent Bernoulli trials, with n trials and success probability p, where the number of heads x follows a binomial distribution and the PMF is defined.

Explore how to implement a binomial trial in python, using numpy vectorization to simulate bernoulli trials, analyze distributions with histograms, and observe how n and p shape the binomial distribution.

Explore how real data sets reveal discrete and continuous random variables, from iris features to Titanic attributes, and learn to model conditional and joint distributions, pmf, for classification and regression.

Explore famous discrete random variables beyond Bernoulli, binomial, and geometric, and complete exercise 01 by proposing another discrete random variable not yet discussed in the course.

Explore the Poisson random variable, its probability mass function and lambda parameter, and learn how Poisson connects to binomial and is approximated by Gaussian distributions, with applications across domains.

Solve a homework problem on the maximum of three rolls of a four-sided die, assuming a uniform PMF, and find the probability that the maximum is even.

Explain how continuous random variables have uncountable values, assign zero probability to individual outcomes, and use interval probabilities and density functions to enforce normalization.

Assess whether the midpoint of a randomly selected interval from a 0–10 line yields a continuous or discrete random variable. Six disjoint intervals have known lengths.

Explain why the midpoint of a randomly selected among six disjoint intervals yields a discrete random variable, with a finite set of midpoints and countable values rather than a continuum.

Explore probability density functions for continuous variables, assigning probabilities to intervals via area under the curve; unlike probability mass functions, the height may exceed one, and total area equals one.

Identify the key properties of a valid probability density function for a continuous random variable x and understand how these properties govern its behavior.

Identify properties of a valid probability density function for a continuous random variable: non-negativity, a normalization condition with area under the curve equal to one, and no restriction on x.

Master the uniform distribution as a continuous random variable on [a, b], with density 1/(b-a) and zero outside, and compute probabilities as area under the curve using NumPy.

Evaluate whether the die roll X forms a uniform random variable by confirming equal probability for outcomes 1–6. Illustrate how discrete uniform variables arise from fair dice in data modeling.

Explains that a fair die yields a discrete uniform random variable, with pmf of 1/6 for each outcome from 1 to 6, illustrating that uniform variables exist in discrete form.

Generate uniform random numbers with numpy, scale from 0–1 to 0–100 and shift to 20–120, and visualize with histograms and KDE plots to illustrate uniformity.

Explore the exponential continuous random variable with non-negative values, its density f(x) = lambda e^{-lambda x} for x >= 0, and how the rate parameter lambda shapes the distribution.

Explore how changing lambda, the arrival rate parameter, affects the exponential distribution, comparing larger versus smaller values and their impact on the density function.

The lecture shows how varying lambda shapes the exponential distribution: larger lambda accelerates decay and increases the peak, while smaller lambda slows decay and lowers the peak.

Plot the exponential distribution's density for varying lambda, observe how larger lambda sharpens decay, and note that the area under the curve stays one before the Gaussian distribution is introduced.

Explore the Gaussian (normal) random variable, a continuous real-valued variable. Examine its density function with mean mu and scale sigma, whose peak, decay, and normalization shape the distribution.

Explore how sigma affects a Gaussian distribution, comparing the impact of large versus small sigma, with mu as the other parameter.

Explore how sigma shapes Gaussian random variables, showing that large sigma flattens the bell curve and increases its fatness, while small variance yields a sharper peak and quicker decay.

Explore Gaussian random variables by generating data with mu and sigma in NumPy, visualize distributions with histograms, and understand how changing parameters shifts the curve and shapes the spread.

Understand how continuous and discrete random variables transform into new features, altering pdfs and pmfs, to support classification models and dimensionality reduction.

Explore the cumulative distribution function (CDF), its usefulness, and its relation to discrete and continuous random variables, as you complete a homework task researching CDF online.

Explore the definition of expectation, showing how the mean of a discrete variable uses summation and a continuous variable uses integration, with Bernoulli examples and Python demos.

Examine how the mean from Bernoulli, geometric, and binomial data relates to distribution parameters such as probability of success, and illustrate the law of large numbers with large-sample averages.

The law of large numbers is presented, showing that for iid samples the expected value equals the sample mean as data grows, with Python demonstrations and estimation applications.

Explore the law of large numbers, iid samples, and how the sample mean converges to the population mean, with examples from Bernoulli, binomial, geometric, Poisson, normal distributions and Python demonstrations.

Explore the law of large numbers in Python by simulating i.i.d. samples from Bernoulli, geometric, binomial, and normal distributions to show the sample mean converges to the true mean.

Compute the expected value and moments of transformed random variables using the original pmf or pdf, with examples like y = 3x^2 + 9.

Solve the homework by computing the expected value, variance, and the fourth moment of the maximum of three rolls of a four-sided die, assuming a uniform PMF for the maximum.

Build a Bayes classifier from scratch using iris data, modeling petal length with gaussian distributions, estimating mu and sigma, and using priors and class-conditional densities to predict species.

Explore joint distributions for multiple random variables, including joint pmf and density, marginal distributions, conditioning and expectations, and the law of total probability leading to multivariate Gaussian distribution.

Derive the expectation of the sum of two discrete random variables x and y from their joint pmf, then extend to continuous distributions by replacing sums with integrals.

Derive the expected value of z = x + y for two discrete variables, using the joint pmf and its marginals, and show E[z] = E[x] + E[y].

Derive the expectation of a binomial random variable with parameters n and p, using independent Bernoulli trials with success probability p.

Represent a binomial variable as the sum of n independent Bernoulli(p) trials and use linearity of expectation to find its mean. The result is E[X] = np.

Derive the expectation formula for the product of two independent discrete random variables, X and Y, using their joint pmf to compute E[XY].

Learn that for independent random variables x and y, the expected value of their product equals the product of their expectations, via the discrete joint distribution factorization.

Explore the multivariate Gaussian as a key joint distribution for random vectors, detailing its density, mean vector mu, and covariance matrix (positive definite) in data science and machine learning.

Explore conditioning in random variables, including discrete and continuous cases, using joint and marginal distributions and independence. Learn about conditional independence and naive Bayes.

Explore the classification problem with many random variables and a discrete class; build p(y|x), compare generative vs discriminative modeling, and note neural networks' focus on discriminative approaches.

Learn how Naive Bayes uses the conditional independence assumption to model y given x1 and x2, estimate feature density, and multiply them for the joint density, with text mining applications.

Predict continuous y from multiple random variables using regression based on the conditional density and its expected value, with linear and ridge models, Bayes principles, and the curse of dimensionality.

Examine the curse of dimensionality as more random variables require vast data for reliable joint distributions, and see how histograms with bins reveal probabilities, with PCA as a reduction option.

Implement from-scratch Naive Bayes classifier in Python using iris data from seaborn; assume independence, model each feature's distribution given class, multiply to form the joint, report results with train-test split.

Explore estimation as a key technique for building probabilistic models by understanding parametric distributions, their parameters, and when to apply kernel density estimates for non-parametric data.

Learn how maximum likelihood estimation identifies parameters of a parametric distribution from iid samples by maximizing the joint probability and noting that MLE minimizes KL divergence.

Learn to estimate lambda for exponential data using maximum likelihood and the log likelihood function, deriving lambda = n over s from iid samples.

Explore the maximum a posteriori estimator, treating parameters as random, contrast with MLE, and see how regularization links MAP to generalization, e.g., exponential distribution with lambda.

Learn how logistic regression turns a Bernoulli maximum likelihood model into a powerful binary classifier using a logistic function, leading to binary cross-entropy loss.

Discover ridge regression as a powerful regression model with a ridge regularizer, derived from MAP estimates under Gaussian assumptions and linked to MLE and cross-entropy losses.

Explain how deep neural networks model probability distributions for classification or regression through layered parameters. Show that learning these parameters yields a probability model estimating y from data.

Explore counting principles in combinatorics essential to probability and statistics, derive permutations of n distinct objects, and show there are n factorial total arrangements by sequentially filling n positions.

Explore counting arrangements and combinations without repetition using n factorial, nPk, and nCk, with examples on three objects and bit strings.

Explore the binomial random variable by deriving its pmf from combinations and Bernoulli trials, using n choose k to count k successes in n independent trials.

Derive logistic regression from maximum likelihood estimation for Bernoulli data, modeling the probability of success with a sigmoid function of the extended feature vector x hat, and optimize w.

Explore logistic regression derivation through maximum likelihood with a Bernoulli model, using sigmoid probabilities, log-likelihood, and cross-entropy loss, then apply gradient descent to find w.

Meet instructor Shahzeb Hamid, introduce who I am, define what I Sciences is, and outline the course descriptive outline for statistical modeling explained in Python.

Explore ai sciences, a community of PhDs and ai practitioners on Udemy, teaching machine learning, artificial intelligence, statistics, and data science to beginners, with professional grooming and problem solving.

Explore summary statistics, including mean, median, mode, case studies, standard deviation, variance, and interquartile range, with mathematical formulas and Python hands-on, then learn hypothesis testing, correlation, and regression using sklearn.

Explore data summarization and key statistical terms in Python, including mean, mode, median, standard deviation, variance, and the interquartile range, with calculation methods and a module overview.

Explore summary statistics, quick summaries, and data properties, and compare mean, mode, and median; analyze dispersion with standard deviation, variance, and interquartile range.

Explore summary statistics to quickly summarize data and enable comparisons. Identify mean, median, mode, and IQR as core concepts under this umbrella.

Explore summary statistics by examining different average types: mode, mean, and median, and learn how to calculate and apply them to identify the middle point in data.

Discover how the arithmetic mean is computed by dividing the sum of data points by their count, with examples showing a mean of 6 for 2, 4, 6, 8, 10.

Discover how to compute the median by ordering data and handling odd and even samples, using n+1 over 2 for odd sizes and averaging two middle values for even sizes.

Learn how to find the median by sorting data in ascending order, counting n, and applying the (n+1)/2 formula, illustrated with a three-sample example where the median is 55.

Identify the mode as the most frequent value and classify it into bimodal, trimodal, and multimodal modes; examples include 245567 and 122232 5557 888 showing modes 2, 5, and 8.

Explore a case study using 20 autistic children's data to explain median, mean, and mode, compute the ballpark and exact medians, the column means, and note a trimodal mode.

Explore how to compute the interquartile range (IQR) by splitting data into quartiles, determining Q1 and Q3, and using Q3 minus Q1, illustrated with an age dataset.

Compute variance as the expected squared deviation from the mean, equal to the square of the standard deviation, by dividing the squared deviations by n or n minus one.

Explore standard deviation as a measure of dispersion around the mean, tying it to variance and data distribution. Learn the step-by-step calculation and apply it in Python with hands-on practice.

Explore descriptive statistics in Python by computing mean, mode, median, standard deviation, variance, and iqr using built-in functions and the statistics module in a hands-on Jupyter notebook.

Learn to compute range, standard deviation, and variance in Python using the statistics module, and explore quantiles and a small box plot to visualize data.

Compute quantiles using Python's statistics.quantiles to obtain Q1, Q2, and Q3, then calculate the IQR as Q3 minus Q1 to assess data dispersion.

Explore hypothesis testing fundamentals—null and alternate hypotheses, p-values, test statistics, and critical values—and implement two-sided, left-tail, and right-tail tests in Python.

Explore how hypothesis testing assesses population parameter assumptions, gauges plausibility of a hypothesis, and decides to accept or reject based on data, then review terminology.

Explore hypothesis testing terminologies, including the null and alternate hypotheses, level of significance, critical values, test statistics, z value, and p value, via graphical explanations and Python CDF demonstrations.

Explain the null hypothesis as a population statement assumed true unless evidence contradicts it. Use a textbook price example to show rejecting null supports the alternative hypothesis.

Explore the alternate hypothesis ha as the statement opposite the null, accepted only when evidence supports it, illustrated by mu not equal to eight in a fat-per-serving quality check.

Explore how the test statistic measures data against the null hypothesis and informs the p-value, using the t statistic formula with sample proportion, total sample, and standard deviation.

Explain the p-value as the probability that the observed statistic is as extreme as under the null, guiding whether to reject the null in favor of the alternative.

Explain how two-sided hypothesis tests use two critical values and alpha linked to the significance level to determine null hypothesis rejection. Describe left- and right-tailed tests and their rejection areas.

Identify the level of significance, or alpha, as the threshold for rejecting hypotheses in one- and two-sided tests; apply Python terms like CDF and PSF to calculate hypothesis testing.

compare cessation rates between nicotine patch and placebo, formulate null and alternate hypotheses, report p-values, and conclude nicotine patch yields higher quit rates in the study.

Test the null hypothesis that 50% would say yes, against the alternative that fewer than 50% do, using a 0.45 sample proportion and p-value 0.0116.

Explore a python-based approach to hypothesis testing, introducing pdf, cdf, sf, and their inverses (ppf, isf), and define z as the test statistic with p-value computed via the survival function.

Choose data and state the null and alternative hypotheses. Select and calculate the z value, then make a statistical decision using the p value and level of significance.

This lecture uses Python to perform hypothesis testing on bottle volumes, testing whether the mean differs from 150 cc, with alpha 0.05 and z from observed and population means.

Import numpy as np, pandas as pd, matplotlib.pyplot as plt, seaborn as sns to compute z from (x-mu) over standard error, using standard deviation and sqrt(4) for four bottles, p-value.

Explore how to use the SciPy norm distribution to compute CDF, SF, PPF, and ISF, understand their relationships, and derive critical values.

Calculate z critical values and p values using the standard normal survival function in Python, then decide null hypothesis rejection at alpha 0.05.

Explore correlation and regression in Python, starting with covariance and auto correlation, then learn to test for correlation and interpret linear regression coefficients and visual representations.

Learn how two random variables move together through covariance and correlation, define means of x and y, compute deviation products, and relate covariance to variance while highlighting interpretation and steps.

Learn how the correlation coefficient r measures the strength and direction of a linear relationship between two variables, ranging from -1 to 1, with interpretation and significance testing.

Explore correlation and simple linear regression by modeling revenue as a function of budget, with intercept and slope, and compute slope from covariance over variance, then implement in Python.

Demonstrates linear regression in Python with pandas and numpy, computes covariance and correlation between budget and revenue, and uses describe, mean, and variance on csv data, parsing release dates.

Explore linear regression with scikit-learn by modeling revenue as a function of movie budget, including defining the intercept and slope and evaluating fit with mean squared error and R2 score.

Build a linear regression model with a 70/30 train-test split, predict revenue, and evaluate with mean squared error and R2, then visualize the regression line.

Explore multiple regression through a module overview that visualizes the concept, explains why we need it, and walks through the model, its formula, and implications in statistical modeling with Python.

Explore why multiple regression extends simple linear regression by showing how money depends on multiple factors like experience, time, and degree, and how to fit a regression model.

Learn the multiple regression formula that explains a dependent variable y using multiple independent variables, with intercept, slope for each predictor, error, and the concept of predictors.

Install Power BI, import data, and develop a financial analysis case study with sales analytics, a profit and loss statement, and cross-country comparisons using visuals in Power BI.

Drop the revenue column to form X and Y, train-test split in python, and fit a multiple linear regression model; evaluate with mean squared error, mean absolute error, and R2.