Statistics for Finance

Explore why returns are the fundamental form of financial data, and learn to compute gross and net returns, multi-period returns, and continuously compounded (logarithmic) returns.

Apply independence to coin tosses and die rolls, deriving the distribution of heads in three tosses and the probability the first die exceeds the second (5/12).

Explore the discrete uniform distribution, its constant probability over a finite set, how to compute the constant via normalization, and practical probability calculations using cardinality.

Explore the binomial distribution through sample calculations, including sampling with replacement (p=0.3, n=10) to compute P(X>3) via the complement, and a quality-control example with 50 parts and 0.04 defect rate.

Explain how independence extends from events to random variables, and define independent random variables; apply the product rule to discrete cases and compute sums like x plus y.

Compute the expected value and variance of continuous random variables using pdfs and integrals, applying linearity and independence with practical examples.

Explore discrete bivariate distributions by building and interpreting the joint distribution and joint pmf, examine independence, and practice reading probabilities from two way tables.

Master discrete conditional distributions by deriving the conditional mass function from a joint distribution, using f(x|y)=f(x,y)/f_Y(y). Learn through step-by-step examples and data conversions.

Explore the chi-square distribution as a gamma special case with its link to the normal. Learn its pdf, mean, variance, mgf, degrees of freedom, and sums of chi-squares for sampling.

Explore transformations of discrete random variables, derive the distribution of Y = H(X) using 1-to-1 and non-1-to-1 cases, and illustrate with examples like binomial and uniform X.

Explore skewness, the third standardized moment, as a measure of asymmetry in return distributions, illustrating positive and negative skewness and the need for broader finance distributions beyond normal.

Random samples are independent, identically distributed observations, and the sample mean estimates mu with variance sigma^2/n, linking sampling with replacement, coin tossing, and Bernoulli models to inference.

Explore the sample mean and the sample variance as unbiased estimators of the population mean and variance, their independence under normal sampling, and the chi-square link for inference.

Explore the autocovariance and autocorrelation functions of stationary time series, defining gamma_x(h) and rho_x(h), and illustrate with white noise, ma1, and ar1 models.

Explore market efficiency, the efficient markets hypothesis, and how prices absorb information; learn martingales, discounted prices, and the random walk path for asset prices.

Explore random walk tests for asset prices by using autocorrelation, Ljung-Box, McLeod-Lee, and turning point tests on log returns and squared returns from the mid 2000s S&P 500.

Explore parametric estimation by selecting finite-parameter model families and finding the parameter values that best explain data, using estimators like the sample mean and the method of moments.

Explore maximum likelihood estimation with one-parameter examples, deriving lambda hat as 1 over sample mean for an exponential model and p hat as s over n for a binomial proportion.

Explore maximum likelihood estimation for normal models, deriving mu hat equals x bar and sigma squared hat equals (1/n) sum (xi minus x bar)^2, with one- and two-parameter cases.

This lecture introduces the method of moments as a simpler alternative to maximum likelihood, illustrating moment matching with a one-parameter model and a two-parameter normal model.

Explore one-sided confidence bounds for normal means and derive delta minus and delta plus from x-bar, known sigma, and z-quantiles. Learn to bound mu with a chosen confidence level.

Explore the formal framework of hypothesis testing in statistics, distinguishing null and alternative hypotheses, simple and composite forms, types I and II errors, and how data inform parameter inference.

Learn how test statistics and p-values decide whether to reject the null by comparing data to its null distribution, using x-bar and z-standardized statistics.

Assess the white noise hypothesis for financial returns, considering weak and strong forms. Analyze mid eighties S&P 500 returns using autocorrelation, confidence bands, and lung box and McLeod Lee tests.

Learn exploratory data analysis through visualization, focusing on histograms as a robust univariate tool to visualize data, estimate the empirical density, and understand marginal distributions.

Analyzing the normality of S&P 500 returns across the mid-1990s and mid-2000s, the lecture uses graphical tests and the Jack Barrett test, showing greater deviations in the mid-1990s.