The table given below shows the time to death $Y($in weeks$)$ and the white blood cell count $x$

$($in thousands$)$ for a group of leukaemia patients. Interest lies in the estimation of exponential

survival probabilities based on time to death data $Y,$ taking into account of the effect of the

explanatory variable $x.$

Assume that the time to death variable recorded above can be modelled by an Exponential

distribution with parameter ${}_i={}_0+{}_1{x}_i($Exponential linear regression$).$

$($a$)$ Write down the likelihood function for this data $($in terms of $x_i,y_i)$ and obtain

the maximum likelihood equations for the two parameters ${}_0$ and ${}_1.$

$($b$)$ Using the software R $($or R Studio$),$ optimise the likelihood function of the

given data numerically to obtain the maximum likelihood estimates of the

parameters.

$($c$)$ Report on your results, plot the data and draw conclusions from your analysis.

$($d$)$ Using your fitted model estimate the expected survival times of patients whose

WBC were $20,000.$

$($e$)$ Based on your fitted model, for patients with a WBC of $30,000,$ calculate the

exponential survival probabilities of

surviving $10$ weeks or longer

surviving $20$ weeks or longer, given that they have survived first $10$ weeks.