Question $2$

Calculate the mean $E[Y_t],$ the variance Var$(Y_t),$ the autocovariance ${}_h$ and the autocorrelation function ${}_h$ for the time series below. Assume ${}_t$ is white noise distributed as $N(0,{}^2).$

$($a$)Y_t=Y_{t-1}+{}_t$

$($b$)Y_t={}_t-{}_{t-1}+{}_{t-2}$

$($c$)Y_t=-{\displaystyle \underset{j=1}{\overset{t}{}}}{}_j$

Question $3$

The USgas package contains data on the demand for natural gas in the US$.$ Install and load the USgas package using the code below.

install. packages$("$USgas$")$

library$($USgas$)$

$($a$)$ Create a tsibble from us$\_$total with year as the index and state as the key.

$($b$)$ Plot the annual natural gas consumption by state for the New England area $($comprising the states of Maine, Vermont, New Hampshire, Massachusetts, Connecticut and Rhode Island$).$

Question $4$

$($a$)$ Download tourism.xlsx from Moodle read it into R using readxl: : read$\_$ecel$().$