Homework 6 _SOLUTION

1. For problem 2 in homework 5, find the standard deviation of her profit.2. For problem 6 in homework 5, find the standard deviation of Y . 3. Walpole 6.1. Copied here: Given a continuous uniform distribution on the range [A,B], show that (a) µ = A+B 2 ; (b) σ2 = (B−A)2 12 . 4. In a type of non-destructive testing of a material, photons are shot at the material, and a sensor on the opposite side counts random variable K, the number of photons that make it through. If the material is type A, the probability of a photon making it through are 0.2; if the material is type B, the probability is 0.1. For this test, you don’t know what material it is. You shoot 15 photons at the material and two photons make it through. (a) If it is in fact material A, what is the probability of your measurement (2 out of 15 make it through)? (b) If it is in fact material B, what is the probability of your measurement (2 out of 15 make it through)? (c) If it is in fact material B, what is the mean and variance of K? (You don’t need to derive it, you can use the formula.) 5. In one week of the baseball season, assume a player will bat 20 times. His historical probability of hitting a home run any one time he is at bat is 0.05. What is the probability he will hit four or more home runs in one week? (Hint: This is one minus the probability of 0, 1, 2, or 3 home runs.) 6. Find the following probabilities: (a) Find P [Z < −1.0] for Z a standard normal r.v. (b) Find P [Z 2.54] for Z a standard normal r.v. (c) Find P [−2 ≤ Z ≤ 2] for Z a standard normal r.v. (d) Find P [X < 20] for X a Gaussian r.v. with mean 20 and standard deviation 10. (e) Find P [X < 0] for X a Gaussian r.v. with mean 20 and standard deviation 10. (f) Find P [−5 ≤ X < 5] for X a Gaussian r.v. with mean 5 and variance 20. (g) Find P [−12.5 ≤ X < −6.5] for X a Gaussian r.v. with mean 0 and standard deviation 6.5. 7. A building near the ocean will flood during a hurricane if the storm surge is more than 20 feet. Assume that a hurricane’s storm surge is normal with mean 10 feet and standard deviation 4 feet. What is the probability that the building will flood during a hurricane in this model?

8. Your job is to make sure that a washing machine lasts more than 10 years, the warranty period, with probability 0.99. As it is, the mean lifetime is 10 years with standard deviation 2.5 years, and it is Gaussian distributed. You can leave the standard deviation the same for no cost. To reduce the standard deviation, it costs $100 to reduce the standard deviation to 1.5 years, or $200 to reduce it to 0.5 years. It costs $50 per year (for any positive real value of years) you wish to increase the mean of the lifetime. (For example, if you want the machine to have a mean lifetime of 11.5 years with a standard deviation of 1.5 years, the cost would be $100(11.5 − 10) = $150, plus $100 to reduce the standard deviation to 1.5, for a total cost of $250. However, this configuration would not have probability 0.99 of having a lifetime greater than 10 years.) Find the lowest cost way to achieve the engineering design goals.