22.4UC2 Assignment 1

This assignment is due at my office, office 3.20 at the Department of Computing and Electrical Engineering, on Friday, February 14, before 4:30 pm. Complete the details below and staple this sheet to the front of your solutions. Please underline your family name.

Name: Student number:

Lecturer: Yvan Petillot

1. A module consists of 8 topics. The exam is set to contain questions from two topics, and the students pick one topic and answer those questions.

How many topics should a student prepare to be 75% confident that one of the topic (s)he prepared is covered on the exam? What assumptions did you make for your calculations? (Note: I do not advise to do this in practice.)

2. In proof testing of circuit boards the probability that any particular diode will fail is 0.1%. Suppose that a circuit board contains 200 diodes.

(a) What kind of distribution models the random variable ‘failure’. Motivate your answer.

(b) How many diodes would you expect to fail, and what is the standard deviation of the number that are expected to fail?

(c) What is the probability that at least three diodes will fail on a board.

(d) If four board are shipped to a particular customer, what is the probability that at least three of them will work properly (i.e., that all diodes on these boards will work)?

3. The air pressure in a randomly selected tire put on a certain model new car is normally distributed with mean value 1.9 bar and standard deviation .05 bar.

(a) What is the probability that the pressure for a randomly selected tire exceeds 1.8 bar?

(b) A tire is classed as under-inflated if its pressure is less than 1.7 bar. What is the probability that at least one of the four tires of a new car of this model are under-inflated?

4. A resistor of a certain type is subjected to an accelerated life text, the lifetime has a gamma distribution with mean 22 weeks and standard deviation 10 weeks.

(a) What is the probability that a resistor will last between 10 and 22 weeks?

(b) What is the ninety-fifth percentile of the lifetime distribution?

(c) Suppose the test is aborted after w weeks. What value of w is such that only one-half of 5% of all resistors would still be operating at the end of the test?

5. Let x denote the time of failure (in years) of a component. Suppose that the probability density function of x is

ρ(x) = \frac{32}{(x + 4)^{3}} for x > 0 .

(a) Show that ρ is a density function.

(b) Determine the cumulative distribution.

(c) Calculate the probability that a component will last between 2 and 6 years.

(d) If the component has a salvage value equal to \frac{50}{4+x} when its time to failure is x,

what is the expected salvage value?

6. The mode of a continuous distribution is the value that maximises the density function. (a) What is the mode of a normal distribution with parameters μ and σ?

(b) What is the mode of a uniform distribution with parameters a and b?

(c) What is the mode of an exponential distribution with parameter λ?