Page 83 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 61

3-2 PROBABILITY DISTRIBUTIONS AND PROBABILITY MASS FUNCTIONS 61

EXERCISES FOR SECTION 3-1
For each of the following exercises, determine the range (pos- 3-6. The random variable is the moisture content of a lot of
sible values) of the random variable. raw material, measured to the nearest percentage point.
3-1. The random variable is the number of nonconforming 3-7. The random variable is the number of surface flaws in
solder connections on a printed circuit board with 1000 con- a large coil of galvanized steel.
nections. 3-8. The random variable is the number of computer clock
3-2. In a voice communication system with 50 lines, the ran- cycles required to complete a selected arithmetic calculation.
dom variable is the number of lines in use at a particular time. 3-9. An order for an automobile can select the base model or
3-3. An electronic scale that displays weights to the nearest add any number of 15 options. The random variable is the
pound is used to weigh packages. The display shows only five number of options selected in an order.
digits. Any weight greater than the display can indicate is 3-10. Wood paneling can be ordered in thicknesses of 1 8,
shown as 99999. The random variable is the displayed weight. 1 4, or 3 8 inch. The random variable is the total thickness of
3-4. A batch of 500 machined parts contains 10 that do not paneling in two orders.
conform to customer requirements. The random variable is the 3-11. A group of 10,000 people are tested for a gene
number of parts in a sample of 5 parts that do not conform to called Ifi202 that has been found to increase the risk for lupus.
customer requirements. The random variable is the number of people who carry the
3-5. A batch of 500 machined parts contains 10 that do not gene.
conform to customer requirements. Parts are selected succes- 3-12. A software program has 5000 lines of code. The ran-
sively, without replacement, until a nonconforming part is dom variable is the number of lines with a fatal error.
obtained. The random variable is the number of parts selected.

3-2 PROBABILITY DISTRIBUTIONS AND
PROBABILITY MASS FUNCTIONS

Random variables are so important in random experiments that sometimes we essentially ig-
nore the original sample space of the experiment and focus on the probability distribution of
the random variable. For example, in Example 3-1, our analysis might focus exclusively on
the integers {0, 1, . . . , 48} in the range of X. In Example 3-2, we might summarize the ran-
dom experiment in terms of the three possible values of X, namely {0, 1, 2}. In this manner, a
random variable can simplify the description and analysis of a random experiment.
The probability distribution of a random variable X is a description of the probabilities
associated with the possible values of X. For a discrete random variable, the distribution is
often specified by just a list of the possible values along with the probability of each. In some
cases, it is convenient to express the probability in terms of a formula.

EXAMPLE 3-4 There is a chance that a bit transmitted through a digital transmission channel is received in
error. Let X equal the number of bits in error in the next four bits transmitted. The possible val-
ues for X are {0, 1, 2, 3, 4}. Based on a model for the errors that is presented in the following
section, probabilities for these values will be determined. Suppose that the probabilities are

P1X 02 0.6561 P1X 12 0.2916 P1X 22 0.0486
P1X 32 0.0036 P1X 42 0.0001

The probability distribution of X is specified by the possible values along with the probability
of each. A graphical description of the probability distribution of X is shown in Fig. 3-1.

Suppose a loading on a long, thin beam places mass only at discrete points. See Fig. 3-2.
The loading can be described by a function that specifies the mass at each of the discrete
points. Similarly, for a discrete random variable X, its distribution can be described by a func-
tion that specifies the probability at each of the possible discrete values for X.