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MTH2222 Assignment 2辅导

MTH2222 Mathematics of Uncertainty
Sem 1, 2022
Assignment 2
Due onTuesday May 3rd by 5 pm. Submission via Moodle using folderAssignment-
2. MTH2222 students work is assessed on questions 1,2,3,4,5,6. MTH2225 students
work is assessed on questions 2,3,4,5,6,7.
1 The goal of this problem is to show the following. If X and Y are normally
distributed and are uncorrelated, then they might still be dependent! Sup-
pose that X is a standard normal distributions. Let ξ be a random variable,
independent of X, which takes values in {?1, 1}, each with probability 1/2.
(a) Find the distribution of Y = ξX. [2 marks]
(b) Are X and Y independent? Justify your answer. [3 marks]
(c) Are X and Y correlated?Justify your answer. [3 marks]
(d) Is (X, Y ) bivariate normal? Justify your answer. [2 marks]
[10 marks]
2 Suppose that X1, X2 are independent geometric with parameter p, where
p ∈ (0, 1/2]. Find the p which maximises the probability of the event X1 = X2.
[4 marks]
3 Let X be a Poisson with parameter 1. Prove (step by step) that
P(X < 4) =
∫ ∞
1
1
6
x3e?xdx.
[4 marks]
4 Let X be a random variable with MGF
MX(t) = θ
teθt
2
,
for some parameter θ > 0. Find P(X > ln θ). [4 marks]
5 Find the constant c such that
f(x) = ce?x?e
?x
, with x ∈ IR
is a probability density function. [4 marks]
6 Let p1 < p2 < p3 . . . be the prime numbers, i.e. natural numbers which are
not the product of two smaller natural numbers (1 is not prime with this
definition). For all i ∈ IN, let γi = p?2i , and Xi be a random variable taking
values on {0, 1, 2, . . .}, such that
P(Xi = k) = (1? γi)γki .
Assume (Xi)i are independent. Let M =
∏∞
i=1 p
Xi
i . Find the p.m.f. of M . You
might need that
∑∞
k=1 k2 = pi2/6 and that each natural number has a unique
decomposition in terms of products of primes.
[6 marks]
7 For MTH2225 Students only. Let (Sn)n be a simple random walk. Find
the (approximate) probability
P(S10000 ≥ 100).
[4 marks]

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