Please provide all the detail equations.

Part I:You have been asked to investigate defects and flaws in several aspects of jet engine manufacturing. For questions 1-4, briefly describe how you used the data to determine your answer. Then show your solution to the problem. Include the work for any calculations you do. You don't need to do that for question 5just provide responses to the questions.

1. A protective coating is applied to the turbine blades. The thickness of the coating is measured in microns and the measurement device provides only integer values. The thickness measurements are uniformly distributed with values from a minimum of 140microns to a maximum of 165 microns.

a. Determine the probability that the coating thickness on a randomly selected turbine blade is 145 microns or less.

b. Determine the mean and variance of the coating thickness for this process.

2. The number of flaws in metal sheeting manufactured for the engine covers is assumed to be Poisson distributed with a mean of 0.1 flaw per square meter.

a. What is the probability that there is exactly 1 flaw in 10 square meters of metal?

b. What is the probability that there are no flaws in 20 square meters of metal?

3. The thickness of a flange on the engine's outer case is uniformly distributed between 0.95 and 1.08 millimeters.

a. Determine the cumulative distribution function of flange thickness. Include probabilities for results outside this range.

b. Determine the proportion of flanges that have a thickness that exceeds 1.02 mm.

c. Determine the mean and variance of flange thickness.

4. The manufacturing of a semiconductor chip used in the turbine engine produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips.

a. Identify which distribution applies when counting the number of chips that are defective in a lot of 1000, and demonstrate that requirements for using this distribution are met in this situation.

b. Identify which distribution can be used to approximate the value of the distribution from

c. (a) and demonstrate that the requirements for using this distribution are met in this situation.

d. Calculate the probability that more than 22 chips in a lot of 1000 are defective.

e. Calculate the probability that the number of defective chips in a lot of 1000 is from 15 to 25 chips (inclusive).

For (c) and (d), you can use the original distribution or the approximation.

5. Reflect on the use of probability distributions in the questions above.

a. Explain why we use known distributions for these calculations.

b. Explain why different formulas were used for the protective coating of the blades in question (1) and for the thickness of the flange in question (3) when both variables follow a uniform distribution.