Books are packed into cartons. The weight W of a book is a continuous random variable with PDF
\,\displaystyle f_W\left (w\right )=\lambda e^{-\lambda w} \quad w \geq 0The number K of books in any carton is a random variable with the PMF
\,\displaystyle p_K\left (k\right )=\frac{\mu^k}{k !} e^{-\mu} \quad k=0,1,2, \ldotsIf we randomly select a carton and its weight is \,\displaystyle X, determine
a. the s-transform of the PDF of X.
b. E[X].
c. the variance of X.
a. The s-transform of the PDF of W is given by
\,\displaystyle M_W\left (s\right )=\frac{\lambda}{s+\lambda}Similarly, the z-transform of the PMF of K is given by
\,\displaystyle G_K\left (z\right )=e^{\mu\left (z-1\right )}=\exp \left (\mu\left (z-1\right )\right )Thus, the s-transform of the PDF if X is given by
\,\displaystyle M_X\left (s\right )=G_K\left (M_W\left (s\right )\right )=\exp \left (\mu\left\{\frac{\lambda}{s+\lambda}-1\right\}\right )=\exp \left (-\frac{\mu s}{s+\lambda}\right )b. The expected weight of the randomly selected carton is
\,\displaystyle E[X]=E[K] E[W]=\mu\left (\frac{1}{\lambda}\right )=\frac{\mu}{\lambda}c. The variance of X is given by
\,\displaystyle \sigma_X^2=E[K] \sigma_W^2+\{E[W]\}^2 \sigma_K^2=\mu\left (\frac{1}{\lambda^2}\right )+\left (\frac{1}{\lambda^2}\right ) \mu=\frac{2 \mu}{\lambda^2}