By Steve Dobbs, Jane Miller, Julian Gilbey

Written to compare the contents of the Cambridge syllabus. records 2 corresponds to unit S2. It covers the Poisson distribution, linear mixtures of random variables, non-stop random variables, sampling and estimation, and speculation checks.

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**Extra resources for Advanced Level Mathematics: Statistics 2**

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Bull. Amer. Math. Soc. 41: 667–670. 2 General Topology General topology has to do with, among other things, notions of convergence. Given a sequence xn of points in a set X , convergence of xn to a point x can be defined in different ways. One of the main ways is by a metric, or distance d, which is nonnegative and real-valued, with xn → x meaning d(xn , x) → 0. The usual metric for real numbers is d(x, y) = |x − y|. For the usual convergence of real numbers, a function f is called continuous if whenever x n → x in its domain, we have f (xn ) → f (x).

If”: suppose a set B is not open. Then for some x ∈ B, by (b) there is a net / B for all i. xi → x with xi ∈ (d): In the proof of (a) we can take the filter base of neighborhoods N = {y: d(x, y) < 1/n} to get a sequence xn → x. The rest follows. For any topological space (S, T ), a set A ⊂ S is said to be dense in S iff the closure A = S. Then (S, T ) is said to be separable iff S has a countable dense subset. For example, the set Q of all rational numbers is dense in the line R, so R is separable (for the usual metric).

A topological space (S, T ) is said to satisfy the second axiom of countability, or to be second-countable, iff T has a countable base. Clearly any second-countable space is also first-countable. 4. Proposition A metric space (S, d) is second-countable if and only if it is separable. 32 General Topology Proof. Let A be countable and dense in S. Let U be the set of all balls B(x, 1/n) for x in A and n = 1, 2, . . To show that U is a base, let U be any open set and y ∈ U . Then for some m, B(y, 1/m) ⊂ U .