Stat 330 (Spring 2015): Homework 3
Due: February 06, 2015
Show all of your work, and please staple your assignment if you use more than one sheet. Write your name,
the course number and the section on every sheet. Problems marked with * will be graded and one additional
randomly chosen problem will be graded. Show all work for numerical problems for full credit.
1. *
(a) Plants A, B, C produce 35% , 15% and 50% of the total output, respectively. Of the output from
each plant, 75% , 95% and 85%, are nondefective, respectively. A customer receives a defective
product. What is the probability that it came from plant C? Show work.
(b) All athletes at the Olympic games are tested for performance-enhancing steroid drug use. The
imperfect test gives positive results (indicating drug use) for 90% of all steroid-users but also (incorrectly) for 2% of those who do not use steroids. Suppose that 5% of all registered athletes use
steroids. If an athlete is tested negative, what is the probability that he/she uses steroids? Show
work.
2. The probability of being color blind is 0.05 for males and 0.0025 for females. In a class of 40 boys and
10 girls, what is the probability that a randomly chosen student from this group is color blind?
3. * A computer has a dual-core processor. At any time, the probability that each of the processors are
active is
Processor 1
In Use
Not In Use
Processor 2
In Use Not In Use
0.50
0.15
0.25
0.10
0.75
0.25
0.65
0.35
Let A be the event that processor 1 is in use and B be the event that processor 2 is in use.
(a) Calculate P (A|B).
(b) Are the events A and B independent? Why or why not?
(c) Calculate P (B|A)
(d) Demonstrate that P (A|B)P (B) = P (B|A)P (A).
4. Let X be a random variable with image Im(X) = {0, 1, 2, 3}.
(a) Determine the value of a in the table below to make it a valid probability mass function:
x
pX (x)
0
0.5
1
0.25
2
0.1
3
a
(b) Determine the probability that...
i. X is at least 2.
ii. X is neither 0 nor 2.
iii. X is no more than 1.
(c) Find the expected value and variance of X.
(d) Let Y be a random variable with Y = 5 − 2X. Determine the image of Y .
(e) Using the rules for computing expected values of a linear function of a r.v., find the expected value
of Y , using the corresponding values of X.
(f) Derive the cumulative distribution function for X and draw it in a chart.
5. (Baron’s book): 2.23 (Calculate the reliability only for the system shown in Figure 2.8(e))
6. (Baron’s book): 2.24
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