Please provide detailed solutions to the following problems/exercises

1) Find the following probabilities:

a) P (0 ≤ Z ≤ .5)

b) P (Z ≤ .5)

c) P (Z ≤ -.5)

d) P (Z ≥ .5)

e) P (- .5 ≤ Z ≤ .5)

2) Dearborn Tech produces a circuit board that has an average life span of 4,500 hours. The life span is normally distributed with a standard deviation of 500 hours. The firm is considering a 3,800 hours' warranty on the circuit board. If this warranty policy is adopted, what proportion of circuit boards should the firm expect to replace under warranty?

3) Suppose that the distribution of a random variable X is approximately exponential with a mean of 10:

a) What is the P (X ≥ 8)?

b) What is the P (X ≤ 11)?

c) What is the P (6 ≤ X ≤ 10)?

d) What is the standard deviation of X?