MAT013 Coursework

Deadline: April 26, 2019

Instructions

The outputs of this coursework will be:

A written report in doc-, pdf- or html-format describing your code (SAS and R) and screenshots with comments to be handed in to Andrey Pepelyshev by email.
A file containing the required SAS code. Name this file STUDENTNUMBER-SAS-lastname (eg. 123456-SAS-Evans) and email it to Andrey Pepelyshev with MAT013 as the subject. Note that all operations needed to complete the coursework should be included in the SAS code.
A file containing the required R code. Name this file STUDENTNUMBER-R-lastname (eg. 123456-R-Evans) and email it to Andrey Pepelyshev with MAT013 as the subject. Note that all operations needed to complete the coursework should be included in the R code.

Coursework

Using R:

Create a function that will give all numbers less than $k$ (an input), not divisible by $3,7$ or $13$. Furthermore let your function take as input the name of a file and write those numbers to a csv file with that name.

Demonstrate this with $k=2341$ and the file name “classtest”.

[15]
A perfect number is a natural number that is equal to the sum of its divisors (excluding itself). For example $1,2,4,7$ and $14$ divide $28$ and $28=1+2+4+7+14$.

Write code in SAS that allows one to write to a csv file a data set with all natural numbers less than a given parameter $N$ as well as a boolean variable indicating if the number is perfect or not. For example, for $N=6$ the csv file would contain the following:
```
 1, False
 2, False
 3, False
 4, False
 5, False
 6, True
```
[30]
Using R:

Write a function that will return the $n$th Fibonacci number, $F(n)$.

Modify the function so that it returns the $n$th number of the sequence defined by:
$\begin{aligned} K(0)&=a\\ K(1)&=b\\ K(n)&=\alpha K(n-1)+\beta K(n-2)\\ \end{aligned}$
Where $a,b,\alpha$ and $\beta$ are input parameters.

Adapt your function so that it will write all numbers of the form $K(n)$ less than some number $k$ to a csv file. The name of the csv file must not be an input parameter to the function but include the parameters $a,b,\alpha$ and $\beta$ as well as the date on which the code was run. For example: general_fib_a=2_b=3_alpha=10_beta=2_2018-04-24.csv.

[10]
Using R:

Obtain the profits (by computing the difference "costs $-$ income") of each company that is present in both data sets incomes.csv and costs.csv.

Draw a histogram of these profits and also obtain a linear relationship between income and profits.

[25]
Suppose that we want to compute the integral
$I=\int_{0}^{\infty}f(x)p(x)dx,$
where $p(x)$ is the density of the standard lognormal distribution. Consider the sum
$S_N=1/N\sum_{i=1}^N f(z_i),$
where $z_i,i=1,...,N,$ are independent identically distributed random variables with the standard lognormal distribution. By the Central Limit Theorem (CLT), we have $S_N \to I ~~as ~~N\to\infty,$
Thus, $S_N$ is an estimator of $I$.
Consider the function $f(x)=\sin(x^2)/(1+x^2)$. Using R, write a function GetSN with argument $N$ which returns $S_N$ . Write a file with 40 evaluations of the function GetSN for $N=5000$. What can you say statistically about $I$?

[20]