Assume that the number of faults in the roll, say y, follows a Poisson distribution with rate lambda*x, where x is the length of the roll.
Does this mean the pmf is
f(y; lambda*x) = exp(-lambda*x)(lambda*x)^y / y! ?

So if lambda follows a Gamma distribution with shape and scale parameters alpha and beta, how do I go about finding the distribution of y (which follows the Poisson distribution with rate lamba*x) ?

So I have
\(y_i \sim Poisson(\lambda x_i)\) and \(\hat{\lambda}_{MLE}={\bar{y}/\bar{x}}\)

Now I need to find \(var(\lambda)\)
Should I have
\(var(\lambda)=var({\bar{y}/\bar{x}})=(E(\bar{y}))^2 var(1/\bar{x})+(E(1/\bar{x}))^2 var(\bar{y})+var(1/\bar{x})var(\bar{y}\)

or should I have
\(\lambda = E(y_i)/x_i \)
\(var(\lambda)=var({y_i/x_i})=(E(y_i))^2 var(1/x_i)+(E(1/x_i))^2 var(y_i)+var(1/x_i)var(y_i) \)