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continuation in practice it might be of interest to determine the value of a in the previous problem in order to
let y1 y2 be independent random variables with partial sums xn n ge 1a translate the conclusions of the previous
suppose that a random variable x has a continuous distribution with the p d f given in example 2 of this section find
suppose that a point is chosen at random on a slick of unit length and that the stick is broken into two pieces at that
suppose that a particle is released at the origin of the xy-plane and travels into the half-plane where x gt o suppose
let y1 y2 be an adapted sequence and let cnnbspisin r n ge 1a suppose that eyn1fn ynnbspcn compensate suitably to
toss a symmetric coin repeatedly and setsince n - snnbspequals the number of tails in n tosses it follows that xnnbsp
this is an addendum to problem where a stick of length 1 was repeatedly randomly broken in the sense that the remaining
another non-regular martingale let y y1 y2 be independent random variables with common distribution given byand set
show that a predictable martingale is as constant formally show that if xn fn n ge 0 is a martingale such that xnisin
a construct a martingale that converges to -infin as as n rarr infinsums of independent random variables with mean 0
prove that a non-negative uniformly integrable sub martingale that converges to 0 as as n rarr infin is identically
prove the theoremlet un be fn-adapted and integrable and seti xn fn n ge 0 is a martingale iff un fn n ge 0 is a
please check this claimnext a famous maximal inequality due to doob whereas the inequalities so far have provided
the last two proofs show that integrability of y-nbspis irrelevant for the integrability of the stopped sum and that
classify the exponential distribution the uniform distribution and the pareto
check that the exponential and normal distributions belong to the domain of attraction of the gumbel
let x be a stable random variables with index alpha isin 0 2 suppose that y is a coin-tossing random variable py 1 py
question show that the geometric distribution is infinitely divisible by exhibiting the canonical representation of the
let x1 x2 be independent identically distributed random variables which are also independent of n isin polambda show
show that the central limit theorem cannot be extended to convergence in probability that is if x1 x2 are independent
suppose that x and y are independent identically distributed random variables with mean 0 and variance 1 such thatshow
we know or else it is not hard to prove that if x and y are independent standard normal random variables then x y and
consider an experiment with success probability p the entropy isif p is unknown a natural and best estimate after n