Problem 1
Each hour a machinist spends some time X drilling and some time Y milling, with (obviously) 0<X+Y<1. Suppose that the joint density function for X and Y as some constant f(x,y)=c in the triangle 0<x, 0<y, x+y<1, and zero elsewhere.
| a) | What is the value of c? Why? |
| b) | What is the density function f(x) for the amount of time X spent drilling? |
| c) | What is the density function f(z) for the total amount of time Z=X+Y spent drilling and milling? |
| d) | Are X and Y independent? Would it be a good idea or a bad one to use independent random variables to model the amount of time per hour spent on these two activities? |
Problem 2
Twelve highway bridges are all designed by civil engineers, eight with degrees from Duke University and four trained at NC State. After long and illustrious service, four of the bridges appear in danger of failing, summarized as follows:
| Fail | OK | |
|---|---|---|
| Duke | 1 | 7 |
| NCSU | 3 | 1 |
| a) | Give the joint probability mass function p(x,y) for X and Y. |
| b) | Find the means E[X] and E[Y]. |
| c) | Find the expected product E[X*Y]. |
| d) | Are X and Y independent? Why? |
| e) | Is the covariance of X and Y positive, negative, or zero? Describe what this means, in terms of bridge failures. |
Problem 3
Recall the automobile spark-plugs from Homework 3, with lifetime densities f(x)=0.1 exp(-0.1*x), for x>0.
| a) | Let X be the failure time of a spark-plug and let Y be the time you intended to replace the plug, uniformly distributed on the interval (0,1). What is the probability P[X<Y] that the spark-plug fails before its intended replacement? |
| b) | Suppose we decide upon a new replacement strategy: fix a time T and replace all six spark-plugs after T years. How large must T be to ensure that the probability of failure before time T of any spark-plug is less than 1%? |
Problem 4
Problem 6.43, on page 284 of the textbook (Mendenhall & Sincich).
Optional Exercise
Find the probability density function for the fraction of work-time spent drilling, F=X/(X+Y), from Problem 1 above. Hint: First find the distribution function, P[F<s] for each s>0.