MathStat474 - HW 8

Posted on Thu 26 Mar. Due: Thu Apr 2.

Problem 1. (Exercise 6.10.5)

A manufacturing company employs two devices to inspect output for quality control purposes. The first device is able to accurately detect 99.3% of the defective items it receives, whereas the second is able to do so in 99.7% of the cases. Assume that four defective items are produced and sent out for inspection. Let \(X\) and \(Y\) denote the number of items that will be identified as defective by inspecting devices 1 and 2, respectively. Assume that the devices are independent. Determine:

  1. \(f_{XY}(x,y)\)
  2. \(f_X(x)\)
  3. \(E[X]\).

Problem 2. (Ex.6.10.17)

An article in Clinical Infectious Diseases [“Strengthening the Supply of Routinely Administered Vaccines in the United States: Problems and Proposed Solutions” (2006, Vol. 42(3), pp. S97–S103)] reported that recommended vaccines for infants and children were periodically unavailable or in short supply in the United States. Although the number of doses demanded each month is a discrete random variable, the large demands can be approximated with a continuous probability distribution. Suppose that the monthly demands for two of those vaccines, namely measles–mumps–rubella (MMR) and varicella (for chickenpox), are independently, normally distributed with means of 1.1 and 0.55 million doses and standard deviations of 0.3 and 0.1 million doses, respectively. Also suppose that the inventory levels at the beginning of a given month for MMR and varicella vaccines are 1.2 and 0.6 million doses, respectively.

  1. What is the probability that there is no shortage of either vaccine in a month without any vaccine production?
  2. To what should inventory levels be set so that the probability is 90% that there is no shortage of either vaccine in a month without production? Can there be more than one answer? Explain.

Problem 3. (Ex. 6.10.18)

The systolic and diastolic blood pressure values (mm Hg) are the pressures when the heart muscle contracts and relaxes (denoted as \(X\) and \(Y\), respectively). Over a collection of individuals, the distribution of diastolic pressure is normal with mean 73 and standard deviation 8. The systolic pressure is conditionally normally distributed with mean \(1.6x\) when \(X=x\) and standard deviation of 10. Determine the following:

  1. Conditional probability density function of \(Y\) given \(X=73\)
  2. \(P(Y<115|X=73)\)
  3. \(E[Y|X=73]\)
  4. Recognize the distribution \(f_{XY}(x,y)\) and identify the mean and variance of \(Y\).

Problem 4 — optional! — extra credit.

Exercise 6.10.23.