Chapter Review 545
Define the random variable X to be the price of the lot for 601, 02. (07. Find the
expected value of X.
- Let a random variable X have probability density function as follows:
x 3 4 5
p(X = x) 0.3 0.4 0.3
Compute the variance and standard deviation of X.
- A computer store has purchased three computers of a certain type at $500 apiece. The
computers then are sold for $1000 each. The manufacturer has agreed to repurchase
any computers that remain unsold after one month for $200 each. Let X be the ran-
dom variable that denotes the number of computers sold. Suppose the probabilities
for selling i computers for i = 0, 1, 2, 3 are p(O) = 0.1; p(l) = 0.2; p(2) = 0.3; and
p(3) = 0.4. Let h(x) denote the profit from selling X units. Find the expected value
of h as well as the standard deviation and the variance.
- Student workers find that 75% of all help desk inquiries involve programs with syntax
errors. Let X be the random variable that counts the number of programs with syntax
errors in 10 randomly chosen consultations. Find the expected value, the variance, and
the standard deviation of this random variable.
8.11.4 Using Discrete Mathematics in Computer Science
- Consider sending a job through the series-parallel system of processors as shown:
The system consists of four stages connected in sequence. Each stage i consists of
identical processors connected in parallel. Each processor at stage i has probability pi
of being up. A job makes it through the system if and only if at least one processor at
each stage is up. We want to determine the probability that a job makes it through the
system.
(a) Model this situation with a cross product sample space Q = Q1 X Q2 X 23 X
Q24. Describe in detail the component sample spaces £i, their probability density
functions fi, and the cross product probability density f. (Hint: It may be useful
to assign labels to the individual processors.)
(b) Let Ai C "i be the event "at least one of the processors at stage i is up." List the
elements of Q2i belonging to Ai. What is P(Ai)?
(c) Let Bi C Qi be the event "none of the processors at stage i is up." List the elements
of Bi. What is P(Bi)?
(d) Describe the event A. c £i that corresponds to Ai g Qi. What is P(A*)?
(e) Describe the event Bi C £2 that corresponds to Bi c Q2. What is P(B7)?
