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1. Find the cumulative distribution function and instantaneous hazard rate function for the following three cases: (10 Points)

a. Case 1: PDF: f(t) = *e-t where t >=0 and >=0 b. Case 2: PDF: f(t) = (2/n) * t* e[(-t^2)/n] where t >=0 and n >=0

2. The weight, X, of a metal ingot produced in a casting foundry follows normal distribution. The mean and standard deviation of X are = 520 pounds and = 11 pounds respectively. Find the following: (10 points)

a. P(X

4. A local manufacturer of toaster includes a repair warranty with the product. The toaster exhibits an exponential increasing failure rate, approximated by the function: r(t) = 2.8 t1.4, where t is measured in years of continuous operation. What is the probability that the time between two successive failures exceeds three months of operation? (20)

5. The reliability of a forging press is given by

R(t) = (1-t/Tmax)^3 for 0