One piece of pvc pipe is to be inserted inside another piece. the length of the first piece is normally distributed with mean value 19 in. and standard deviation 0.6 in. the length of the second piece is a normal random variable with mean and standard deviation 17 in. and 0.5 in., respectively. the amount of overlap is normally distributed with mean value 1 in. and standard deviation 0.2 in. assuming that the lengths and amount of overlap are independent of one another, what is the probability that the total length after insertion is between 32.65 in. and 35.35 i g

Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?