Home Page > > Details

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]

Contact Us(Ghostwriter Service)

- QQ：99515681
- WeChat：codinghelp
- Email：99515681@qq.com
- Work Time：8:00-23:00

- Programhelp With ,Help With C++ Course... 2022-05-10
- Help With Data Programming,Help With C... 2022-05-10
- 5Cce2sashelp With ,Python，Java Progra... 2022-05-10
- Help With Program Programming,Help Wit... 2022-05-09
- Help With Csci 3110,Help With Java，Py... 2022-05-09
- Mth2222help With ,Help With C/C++，Pyt... 2022-05-09
- Cse3bdchelp With ,Help With Sql Progra... 2022-05-08
- Help With Cis 468,Help With Java，Pyth... 2022-05-08
- Comp Sci 4094/4194/7094 Assignment 3 D... 2022-05-07
- Cs 178: Machine Learning & Data Mining... 2022-05-07
- Data7703 Assignment 4 2022-05-07
- Data Programminghelp With ,Help With S... 2022-04-25
- Help With Ait681 Course,Help With Pyth... 2022-04-25
- Cse121l Programminghelp With ,Help Wit... 2022-04-25
- Help With Iti1120,Help With Java，C/C+... 2022-04-25
- Cmt304help With ,Help With C++，Python... 2022-04-25
- Help With Engn4528,Matlab Programmingh... 2022-04-24
- Help With Fin 2200,Help With Java，Pyt... 2022-04-24
- Bism 7255Help With ,Help With Java，Py... 2022-04-23
- Comp202help With ,Help With Java Progr... 2022-04-23

Contact Us - Email：99515681@qq.com WeChat：codinghelp

© 2021 www.asgnhelp.com

Programming Assignment Help！