PU-106 · Experimental Physics & Data Analysis I
Starting from the axioms of probability, this unit builds the full machinery a physicist needs to turn noisy measurements into defensible numbers with honest error bars, culminating in maximum-likelihood estimation and least-squares fitting. The intellectual arc runs from "what is a random variable" through the central limit theorem to parameter estimation and goodness-of-fit, so that every later experimental and computational unit inherits a rigorous account of what an uncertainty actually means.
Lectures
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Derivations homed in this unit
Expectation, Variance and Their Algebra
Derive the linearity of expectation and the identity Var(X)=E[X^2]-E[X]^2, and the scaling rules for aX+b, directly from the definition of expectation.
Variance of Linear Combinations and Covariance
Derive Var(sum a_i X_i) = sum a_i a_j Cov(X_i,X_j), reducing to additivity of variances for independent variables.
First-Order Propagation of Uncertainty
Derive the general error-propagation formula for f(x_1,...,x_n) by first-order Taylor expansion and the definition of variance.
Propagation with the Covariance Matrix
Derive the Jacobian form Sigma_y = J Sigma_x J^T for propagating a full covariance matrix through a vector-valued function.
The Standard Error of the Mean
Derive that the sample mean of N independent measurements has variance sigma^2/N, so its uncertainty falls as 1/sqrt(N).
Bessel's Correction for Sample Variance
Show that dividing by N-1 rather than N gives an unbiased estimator of the population variance when the mean is itself estimated from the data.
The Inverse-Variance Weighted Mean
Derive that combining measurements with weights proportional to 1/sigma_i^2 minimises the variance of the combined estimate.
The Binomial Distribution from Bernoulli Trials
Derive the binomial pmf, mean Np, and variance Np(1-p) from N independent identical Bernoulli trials.
Poisson Statistics of Counting
Derive the Poisson distribution as the limit of the binomial for large N and small p at fixed mean, giving the sqrt(N) rule for counting experiments.
Characteristic Functions and Moments
Define the characteristic function E[e^{itX}], and show its derivatives at zero generate the moments and it factorises for sums of independent variables.
The Central Limit Theorem
Prove that the standardised sum of many independent finite-variance variables converges to a Gaussian, via expansion of the characteristic function.
The Method of Maximum Likelihood
Formulate the likelihood of a dataset given parameters and derive the estimating equations by maximising the log-likelihood.
Least Squares from Gaussian Likelihood
Show that maximising the likelihood under independent Gaussian errors is exactly minimising the chi-squared sum of squared residuals.
Linear Least Squares and the Normal Equations
Derive the normal equations (A^T W A) p = A^T W y for the best-fit parameters of a linear model, in matrix form with a weight matrix.
Parameter Uncertainties and the Cramer-Rao Bound
Derive the parameter covariance matrix as the inverse Fisher information / curvature of the log-likelihood, and the Cramer-Rao lower bound on estimator variance.
The Chi-Squared Distribution
Derive the probability density of a sum of k squared standard-normal variables, obtaining the chi-squared distribution with mean k and variance 2k.
Goodness of Fit and Degrees of Freedom
Derive the expected chi-squared of a fit and show each fitted parameter removes one degree of freedom, giving the reduced-chi-squared test.