$[25$ points$]$ The purchase of a new piece of electronic measuring equipment for use in a continuous

metal$-$forming process is being considered. If this equipment were purchased, the capital cost would be

$$418,000,$ and the estimated savings are $$148,000$ per year. The useful life of the equipment in this

application is uncertain. The estimated probabilities of different useful lives occurring are shown in the

following table. Assume that MARR $=15\%$ per year before taxes and the market value at the end of its

useful life is equal to zero. Based on a before$-$tax analysis,

$($a$)$ What are the $E(PW),V(PW),$ and $SD(PW)$ associated with the purchase of the equipment?

$($b$)$ What is the probability that the PW is less than zero?