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A scan statistic searches a family of candidate locations or subsets and reports the largest local statistic. Under a null hypothesis, a union bound converts a tail estimate for each candidate into a bound for the maximum; no independence between local statistics is required. Structured alternatives often permit a chi-squared testing lower bound using the overlap geometry of two candidates.

A source symmetry is a transformation of possible plaintexts that preserves their probability distribution. If different cryptographic keys induce candidate decryptions related by such a symmetry, source statistics cannot distinguish those keys, even when the ciphertext becomes arbitrarily long.

Let independent plaintext bits $X_i$ be fair, and reuse one independent fair key bit $K$ through $C_i=X_i\oplus K$. Flipping every plaintext bit preserves the source law. For either key value, $\mathbb{P}(C_1=c_1,\dots ,C_n=c_n\mid K)=2^{-n}$, so observing any finite ciphertext leaves the two key probabilities equal. This example has only two keys but no finite unicity distance; redundancy assumptions are essential to a source-based threshold estimate.^[ref]

Statistics studies how to learn from data while accounting for variation and uncertainty. Descriptive statistics summarizes observations; statistical inference uses an explicit statistical model to estimate unknown quantities or assess claims. For example, the sample mean estimates a population mean, while its standard error describes variation across hypothetical repeated statistical samples. A large dataset does not automatically remove bias from statistical sampling or confounding. The design generating the observations is part of the reasoning, rather than an optional prelude to calculating a formula.

Thomas S. Ferguson works on statistics, probability theory and game theory. His UCLA page documents research on statistical decisions, stopping problems and games.^[ref]

His 1973 paper introduced a Bayesian framework built on the Dirichlet process, a probability distribution over probability measures. If the prior parameter measure is $\alpha$, observing $x_1,\dots ,x_n$ changes it to $\alpha +{\sum }_i{\delta }_{x_i}$. This conjugate update extends the familiar idea of adding category counts to inference about an entire unknown distribution, providing a foundation for Bayesian nonparametric statistics.^[ref]

Students take three introductory subjects from the published list, including Probability theory and statistics. The branches represent the available subjects, not five compulsory modules.

Students take three introductory subjects. Probability theory and statistics is required to continue with Psychology after the first year.

SB1 applied and computational statistics is compulsory; students must also take at least two units from SB2 and SB3.

Statistics options and a compulsory statistics-project dissertation, with mathematics options available under the course rules.