DescriptionMTH2222/5 Assignment 2

Sem 1, 2023

Due April 28 by 11.55 pm.

Submission via Moodle using folder Assignment- 2. MTH2222 students work is

assessed on questions 1,2,3,4,5,6,7. MTH2225 students work is assessed on questions

1,2,3,4,5,6,7,8. All the solutions to questions below must be justified.

1 Suppose 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.

Define Y = ξX.

(a) Are X and Y independent? Justify your answer. [4 marks]

(b) Are X and Y correlated? Justify your answer. [4 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 .

[5 marks]

3 Let p1 < p2 < p3 . . . be the prime numbers, i.e. natural numbers which are not
the product of two smaller natural numbers (do not consider 1 to be prime).
For all i ∈ IN, let γi = p−2
, and Xi be a random variable taking values on
i
{0, 1, 2, . . .}, such that
P(Xi = k) = (1 − γi )γik .
Xi
. Find the p.m.f. of M . You
Assume (Xi )i are independent. Let M = ∞
i=1 pi P
2
−2
= π6 and that each
might need to use (without a proof) the identity ∞
k=1 k
natural number has a unique decomposition in terms of products of primes.
[10 marks]
Q
4 Let Z be a standard normal random variable, whereas X = 3Z + 2. Find
Var(XZ). [4 marks]
5 Suppose that X is distributed as a Gamma(α, λ), where α > 3.

(a) Find an expression for E[X k ] for all k ∈ IN. [4 marks]

(b) Find an expression for E[X −2 ]. [4 marks]

(c) What is larger between E[X −1 ] and E[X]−1 ? [3 marks]

6 Let X be a random variable with PDF f (x) = c|x|α−1 e−λ|x| , for x ∈ (−∞, ∞),

where α, λ are positive.

(a) Find c. [4 marks]

(b) Identify the conditional distribution of X given X > 0. [4 marks]

(c) Compute E[X n ] for any n ∈ IN. [4 marks]

7 For MTH2225 Students only. Let (Sn )n be a simple random walk, i.e.

P

Sn = ni=1 Xi , with (Xi )i being i.i.d. mean zero random variables each taking

values in {−1, 1}. Find the (approximate) probability

P( max Si ≥ 100).

i≤10000

[10 marks]

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