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PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 66

66 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

3-29. Determine the cumulative distribution function for Determine each of the following probabilities:
the random variable in Exercise 3-19. (a) P1X 42 (b) P1X 72
3-30. Determine the cumulative distribution function for (c) P1X 52 (d) P1X 42
the random variable in Exercise 3-20. (e) P1X 22
3-31. Determine the cumulative distribution function for 3-35. 0 x 10
the random variable in Exercise 3-22. 0.25 10 x 30
F1x2 µ
3-32. Determine the cumulative distribution function for 0.75 30 x 50
the variable in Exercise 3-23. 1 50 x
Verify that the following functions are cumulative distribution (a) P1X 502 (b) P1X 402
functions, and determine the probability mass function and the (c) P140 X 602 (d) P1X 02
requested probabilities. (e) P10 X 102 (f) P1 10 X 102
3-33. 0 x 1 3-36. The thickness of wood paneling (in inches) that a cus-
F1x2 •0.5 1 x 3 tomer orders is a random variable with the following cumula-
1 3 x tive distribution function:
(a) P1X 32 (b) P1X 22
(c) P11 X 22 (d) P1X 22
3-34. Errors in an experimental transmission channel are 0 x 1 8
found when the transmission is checked by a certifier that de- F1x2 µ 0.2 1 8 x 1 4
tects missing pulses. The number of errors found in an eight- 0.9 1 4 x 3 8
bit byte is a random variable with the following distribution: 1 3 8 x

0 x 1 Determine the following probabilities:
0.7 1 x 4 (a) P1X 1 182 (b) P1X 1 42
F1x2 µ
0.9 4 x 7 (c) P1X 5 162 (d) P1X 1 42
1 7 x (e) P1X 1 22

3-4 MEAN AND VARIANCE OF A DISCRETE RANDOM VARIABLE

Two numbers are often used to summarize a probability distribution for a random variable X.
The mean is a measure of the center or middle of the probability distribution, and the variance
is a measure of the dispersion, or variability in the distribution. These two measures do not
uniquely identify a probability distribution. That is, two different distributions can have the
same mean and variance. Still, these measures are simple, useful summaries of the probabil-
ity distribution of X.

Definition
The mean or expected value of the discrete random variable X, denoted as or E1X2, is

E1X2 a xf 1x2 (3-3)
x
The variance of X, denoted as 2 or V1X2, is

2
2
2
2
V1X2 E1X 2 a 1x 2 f 1x2 a x f 1x2 2
x x
The standard deviation of X is 2 2 .