Problem 1
The time required for a technician to machine a specific component of a product is normally distributed with mean of 2 hours and a standard deviation of 17 minutes.
| a) | What is the probability that the technician can machine one component in 1.5 hours or less? |
| b) | What is the probability that the technician will require at least 2.75 hours to complete one component? |
Problem 2
The sum of two independent uniform random variables has a Triangle distribution, with density function f(x)=x, for 0<x<1; f(x)=2-x, for 1<x<2; f(x)=0, for other x.
| a) | What is the probability that the sum is less than 1/2? |
| b) | What is the expectation of the sum? Show your work. |
Problem 3
The simplest probability distribution useful for modeling lifetimes is the exponential distribution, with density function f(x)=exp(-x/m)/m, for x>0, and f(x)=0, for x<0; m is the mean (expectation) of the distribution. Use the Exponential distribution with mean m=10 to model the time-to-failure (in years) for automobile spark-plugs.
| a) | What is the probability that a spark-plug lasts at least its mean, 10 years? |
| b) | Assuming independence, what is the probability that a six-cylinder car (with six spark-plugs) lasts one year without any spark-plug failure? |
Problem 4
Problem 5.12, on page 216 of the textbook (Mendenhall & Sincich).
Optional Exercise
The times required by the technician for different components in Problem 1 above are independent. The sum of independent normally-distributed random variables again has the normal distribution, whose mean is the sum of the means and whose variance (not standard deviation) is the sum of the variances. What is the probability that the technician will be able to complete four components within a 7-hour working day?