Liquid crystal displays (LCDs) that your wholesaling company is marketing for a large Japanese electronics firm are known to have a distribution of lifetimes of the following Gamma-distribution form:

\(f(z ; \alpha)=\frac{1}{2^{\alpha} \Gamma(\alpha)} z^{\alpha-1} e^{-z / 2} I_{(0, \infty)}(z)\), where \(z\) is measured in 1,000 's of hours.

A set of \(n\) iid outcomes of \(Z\) will be used in an attempt to obtain information about the expected value of the lifetime of the LCD's.

(a) Define the functional form of the joint density of the iid random variables say \(\left(X_{1}, \ldots, X_{n}ight)\), of LCD lifetimes.

(b) What is the density function of the random variable \(Y_{n}=\sum_{i=1}^{n} X_{i}\).

(c) Supposing that \(n\) were large, identify an asymptotic distribution for the random variable \(Y_{n}\). (Note: since you don't know \(\alpha\) at this point, your asymptotic distribution will depend on the unknown value of \(\alpha\).)

(d) If \(\alpha\) were equal to \(1 / 2\), and the sample size was \(n=20\), what is the probability that \(y_{n} \leq 31.4104\) ? Compare your answer to the approximate probability obtained using the asymptotic distribution you defined in (c).

(e) If \(\alpha\) were equal to \(1 / 2\), and the sample size was \(n=50\), what is the probability that \(y_{n} \leq 67.5048\) ? Compare your answer to the approximate probability obtained using the asymptotic distribution you defined in (c).