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1 if x y and z are uniform spaces f is uniformly continuous from x into y and g is uniformly continuous from y into
1 a topological space s t is called perfectly normal iff for every closed set f there is a continuous real function f
1 a show that any open set u in r is a union of countably many disjoint open intervals one or two of which may be
1 show that the set rq of irrational numbers with usual topology relative topology from r is topologically complete2
in the proof of proposition 1211 a 1-1 measurable function f from r onto 0 1 was constructed by representing each
let xt be a brownian motion show that for any epsilongt 0 and almost all omega there is an m omega infin such that for
for iid variables x1 x2 let an be the smallest sigma -algebra for which x1 xn are measurable and bn the smallest sigma
1 reconsider the continuous-time markov chain example discussed in class assume now that the machines are not serially
let q run through the collection of all laws for which eq f 2 infin for the u -statistic un show that a un is the
for a given n ge 2 consider the family of binomial distributions for the number k of successes in n independent trials
suppose that y has a poisson distribution py k e-lambdalambdak k k 0 1 suppose y can be observed only when y ge 1
1 the tax rate of 0984 in decimal can be expressed as how many millsa 9840b 9084c 984d 9842 commissions charged on the
the usual sample variance s2 is a u -statistic of order 2 for n 4 x1 x2 - x3 - x424 is another possible estimator of
1 if a car is depreciated in four years what is the rate of depreciation using twice the straight-line ratea 25b 100c
1nbsp let p be a law on a separable normed vector space s middot x isin s and px the translate of p by x so that
1 a 40000 loan at 4 dated june 10 is due to be paid on october 11 calculate the amount of interest assume ordinary
show that there exist three laws alpha beta and gamma on r2 such that there is no law p on r6 with coordinates x y and
1 show that if s is the real line r with usual metric then theorem 1172 holds for the probability space 0 1 with
given a product x iti isini xi of topological spaces xi t with product topology and a directed set j a net in x
4 if x t and y u are topological spaces a is a base for t and b is a base for u show that the collection of all
1 ifnbspsinbspare sets with discrete topologies show that the product topology for finitely many such spaces is also
for any two real numbers u and v maxu v u iff u ge v otherwise maxu v v a metric space s d is called an ultrametric
1 a let q be the set of rational numbers show that the riemann integral of 1q from 0 to 1 is undefined the net in its
1 let x d and y e be pseudometric spaces with topologies td and te metrized by d and e respectively let f be a function
1 on r2 let dx y u v x - u2 y - v212 usual metric ex y x - u y - v show that e is a metric and metrizes the same