The lifetime of a certain computer chip that your company manufactures is characterized by the population distribution

$f(z;\theta )=\frac{1}{\theta }{e}^{-z/\theta }{I}_{(0,\infty )}\left(z\right)$,

where $z$ is measured in thousands of hours. Let $(X_1,\dots ,X_n)$ represent iid random variables with the density $f(z;\theta )$. An outcome of the random variable $Y_n=(1+{\sum }_{i=1}^n{X}_i)/n$ will to be used to provide an estimate of $\theta$.

(a) Is it true that $Y_n\stackrel{m}{\to }\theta$ ? Is it true that $Y_n\stackrel{p}{\to }\theta$ ?

(b) Define an asymptotic distribution for $Y_n$.

(c) Suppose $n=100$ and $\theta =10$. Use the asymptotic distribution you defined in (b) to approximate the probability that $Y_n\ge 15$.