Mathematics for Data Science 101

Understand functions as mapping machines in calculus and data science, illustrated by graphs and the vertical line test, and classify them as piecewise, even, odd, increasing, or decreasing.

Derive the slope of a tangent to y = x^2 at (1,1) using limits, and apply the concept to average and instantaneous velocity.

Explore the derivative as the rate of change of a function, using limits and rise over run to find slopes and tangent lines, with examples on f(x)=x^2-8x+9 and f(x)=3x+2x^4.

Master the rules of differentiation, including power, constant, exponential, product, and quotient rules, to compute derivatives and the slope of tangent lines without manual ab initio limits.

Identify critical points by setting the derivative to zero, then use the first and second derivative tests to classify local extrema and determine global extrema.

The extreme value theorem guarantees a global maximum and minimum for any continuous function on a closed interval, with endpoints included, and it fails if either condition is violated.

Explore inflection points and how to test for concavity changes using first and second derivatives, including inconclusive cases and real-world examples like f(x)=x^3 and f(x)=x^4-2x^3.

Explore random non-deterministic phenomena, define experiments, sample space, sample points, and events, and quantify likelihoods with probability using dice, coins, and cards.

Explore counting techniques to determine sample spaces and probabilities, including the fundamental counting principle, permutations, and combinations, with practical examples like coin flips and license plates.

Understand the three core probability axioms—non-negativity, certainty, and additivity—through a die-roll example, and apply them to unions, intersections, and complements to solve problems.

Explore how Venn diagrams visualize probability, use the addition rule to find unions and intersections, and apply the union formula for two or three not mutually exclusive events.

Apply the total probability theorem to compute event likelihoods by partitioning the sample space into mutually exclusive events, using the law of total probability with intersections.

Apply Bayes theorem to compute posterior probabilities using prior, likelihood, and evidence, with real-world examples like voting demographics and virus tests.

Convert qualitative outcomes into numerical representations through random variables, enabling calculation of probabilities, distributions, mean, and variance. Differentiate discrete and continuous random variables to capture countable vs uncountable value possibilities.

Define and apply the probability mass function for discrete variables, construct it from empirical and theoretical probability, and use the cumulative distribution function through coin flips and practical examples.

Explore probability density function and cumulative distribution function for continuous random variables; learn to compute interval probabilities via integration and why single-point probabilities are zero.

Learn the expected value and variance of discrete and continuous random variables, using sums and integrals of X with the probability mass or density function, including practical sampling problems.

Explore the binomial distribution: fixed number of independent Bernoulli trials with the same success probability. Compute the PMF, mean, and variance to count the number of successes in n trials.

Discover the Poisson distribution: count independent events at a constant mean rate, using the probability mass function with lambda as the mean, with real examples.

Explore how the normal distribution is defined by mean and standard deviation, and use z-scores to locate 68% within one standard deviation, 95% within two, and 99.7% within three.

The central limit theorem shows that with large samples, the distribution of sample means becomes approximately normal, with the mean equal to population mean and standard error sigma over sqrt(n).

Understand population parameters and how sample statistics estimate them, with the population mean as the fixed but unknown parameter and the sample mean as a random variable from sampling variation.

convert ungrouped data to grouped data using Sturges' rule to pick 6 classes, then compute class width from range, and compare inclusive class intervals, boundaries, and exclusive intervals.

Explore measures of dispersion, including range, interquartile range, and variance. Compare standard deviation and coefficient of variation using Messi and Ronaldo's goal data.

Compute variance and standard deviation for grouped data by calculating mean from the frequency distribution, subtracting class marks, squaring differences, weighting by frequency, and using N-1 for sample variance.

Explore null and alternate hypotheses as claims about population parameters, with examples of means and proportions, and identify right-tail, left-tail, and two-tailed tests.

Understand when to reject the null hypothesis in z-tests and t-tests by computing p-values (p < significance) for right, left, and two-tail tests using z-statistic, t-statistic, and cumulative distribution function.

Apply the z-test for mean with known population standard deviation to compute the z statistic and p-value for one- or two-tailed tests, and decide on the null hypothesis.

Apply t-tests for mean with small samples and unknown population standard deviation. Use the t-statistic, degrees of freedom, and one- or two-tailed decisions at set significance levels.

Explore hypothesis testing for the mean of two populations using z-tests and t-tests. Apply standard error calculations, two-tailed and one-tailed tests, and interpret p-values and degrees of freedom.

Use the chi-square test of independence to determine whether two categorical variables are associated in data science contexts, using observed versus expected frequencies and a five percent significance level.