The BigVision Electronic Store sells a large 73 inch diagonal big screen TV. The TV comes with a standard 1 year warranty on parts and labor so that if anything malfunctions on the TV in the first year of ownership, the company repairs or replaces the TV for free. The store also sells an "extended warranty" which a customer can purchase that extends warranty coverage on the TV for another 2 years, for a total of 3 years of coverage. The daily numbers of TVs and extended warranties sold can be viewed as the outcome of a bivariate random variable \((T\), \(W)\) with probability model \((R(T, W), f(t, w))\) given by

\(f(t, w) \begin{cases}(2 t+w) / 100, \text { for } & t \text { and } w=0,1,2,3,4, \text { and } w \leq t \\ 0 & \text { otherwise }\end{cases}\)

a. What is the probability that all of the TVs sold on a given day will be sold with extended warranties?

b. Derive the marginal density function for the number of TVs sold. Use it to define the probability that \(\leq 2\) TVs are sold on a given day?

c. Derive the marginal density function for the number of warranties sold. What is the probability that \(\geq 3\) warranties are sold on a given day?

d. Are \(T\) and \(W\) independent random variables?