5. At an electronics manufacturing plant building circuit boards on an assembly line for customers, employees must insert, solder and check connections on all pins on four different types of boards. The yellow tagged boards require one chip, the blue tagged boards require 1 two chips, the purple tagged boards require three chips and the brown tagged boards require four chips. It has been determined that the time it takes from the moment a board arrives at a work station to insert, solder, and check connections on all pins for one chip is random and is best modeled as an exponential random variable with mean (expected value) of 5 minutes. This is independent of the time it takes to connect a second chip (if any), a third chip (if any), or a fourth chip (if any). All four boards types are produced in large quantities but the probability that a board is yellow tagged, blue tagged, purple tassed, or brown tagged is 0.1, 0.2, 0.3 and 0.4 respectively. This means that an employee that picks up a board for chip insertion may have to insert one, two, three or four chips, depending on the color of the tag the board carries. In order to keep the assembly line moving smoothly, it is important that boards to be worked on not arrive at rates considerably higher than they can be processed and completed, yet not rates that are too slow where employees have idle waiting time for new boards to arrive. Therefore it is important to determine, a) the expected value of the random variable that models the time it takes to go from picking up a board to finishing up the chip insertion task(s), b) the variance of the random variable described in part a), and c) the probability that the random variable described in part a) exceeds the expected value determined in part a). HINT: model as a random sum of random variables. CAUTION: the answer to parte) is NOT one half. 5. At an electronics manufacturing plant building circuit boards on an assembly line for customers, employees must insert, solder and check connections on all pins on four different types of boards. The yellow tagged boards require one chip, the blue tagged boards require 1 two chips, the purple tagged boards require three chips and the brown tagged boards require four chips. It has been determined that the time it takes from the moment a board arrives at a work station to insert, solder, and check connections on all pins for one chip is random and is best modeled as an exponential random variable with mean (expected value) of 5 minutes. This is independent of the time it takes to connect a second chip (if any), a third chip (if any), or a fourth chip (if any). All four boards types are produced in large quantities but the probability that a board is yellow tagged, blue tagged, purple tassed, or brown tagged is 0.1, 0.2, 0.3 and 0.4 respectively. This means that an employee that picks up a board for chip insertion may have to insert one, two, three or four chips, depending on the color of the tag the board carries. In order to keep the assembly line moving smoothly, it is important that boards to be worked on not arrive at rates considerably higher than they can be processed and completed, yet not rates that are too slow where employees have idle waiting time for new boards to arrive. Therefore it is important to determine, a) the expected value of the random variable that models the time it takes to go from picking up a board to finishing up the chip insertion task(s), b) the variance of the random variable described in part a), and c) the probability that the random variable described in part a) exceeds the expected value determined in part a). HINT: model as a random sum of random variables. CAUTION: the answer to parte) is NOT one half