ENGI 3423 Probability and Statistics

Faculty of Engineering and Applied Science
2007 Fall

Term Test 2

2007 October 31
[Single and Joint Probability Distributions, Central Limit Theorem]

1. The joint probability mass function of two discrete random quantities X, Y is given by the table

   p(x, y)		Y
   		-1	0	1
   X	-1	.20	.10	.20
   	0	.08	.06	.08
   	1	.12	.04	.12

   1. Extend the table to display the marginal probability mass functions p_X(x) and p_Y(y).

      [3]

   2. Verify that   p(x, y)   is a valid probability mass function.

      [2]

   3. Show that   E[Y] = 0.

      [2]

   4. Find the covariance of   X and Y.

      [4]

   5. Are X, Y independent?   Why or why not?

      [4]

2. Valves for a prototype machine have lifetimes   T   that are independent random quantities following an exponential distribution with a mean lifetime of 144.27 weeks.

   1. Show that the probability that a randomly chosen valve has a lifetime longer than 200 weeks is .2500, correct to four decimal places.

      [2]

   2. A random sample of 10 such valves is taken.   Let X be the number of valves in the random sample that last longer than 200 weeks.   Show that the probability distribution for X is, (to a good approximation), binomial.

      [2]

   3. Find the probability that more than two of the ten valves in the random sample last longer than 200 weeks.

      [3]

   4. Another random sample of 100 valves is taken.   Find, correct to two significant figures, the probability that the sample mean lifetime exceeds 170 weeks.

      [5]

3. A simple decision tree, together with the relevant probabilities, is shown here.

   

   The consequences (payoffs) are:

   • accept a good item:     + $ 40
   • accept a poor item:     - $ 80
   • reject a good item:     - $   5
   • reject a poor item:         $   0
   The test costs $5.
   1. Determine the optimum strategy and the maximum expected gain.

      [7]

   2. By how much can the test cost be changed without changing the optimum strategy?

      [1]

4. A function   f (x)   of the continuous quantity   x   is defined by
                
   where   a   and   b   are constants.

   1. Find the values or range(s) of values that   a   and   b   must have in order for   f (x)   to be a well-defined probability density function.

      [4]

   2. Show that the corresponding c.d.f. (cumulative distribution function)   F (x)   is
                   

      [4]

   3. Use the c.d.f. in part (b) to evaluate   P[X > 0]   in terms of   a .

      [3]

   4. Find the population mean   µ   in terms of   a .

      [4]