Page 100 - Applied Statistics And Probability For Engineers

PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 78

78 CHAPTER 3 DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

(b) If the manufacturer stocks 102 components, what is the (b) What is the probability the student answers less than 5
probability that the 100 orders can be filled without questions correctly?
reordering components? 3-70. A particularly long traffic light on your morning com-
(c) If the manufacturer stocks 105 components, what is the mute is green 20% of the time that you approach it. Assume
probability that the 100 orders can be filled without that each morning represents an independent trial.
reordering components? (a) Over five mornings, what is the probability that the light is
3-69. A multiple choice test contains 25 questions, each green on exactly one day?
with four answers. Assume a student just guesses on each (b) Over 20 mornings, what is the probability that the light is
question. green on exactly four days?
(a) What is the probability that the student answers more than (c) Over 20 mornings, what is the probability that the light is
20 questions correctly? green on more than four days?

3-7 GEOMETRIC AND NEGATIVE BINOMIAL DISTRIBUTIONS

3-7.1 Geometric Distribution

Consider a random experiment that is closely related to the one used in the definition of a
binomial distribution. Again, assume a series of Bernoulli trials (independent trials with con-
stant probability p of a success on each trial). However, instead of a fixed number of trials,
trials are conducted until a success is obtained. Let the random variable X denote the number
of trials until the first success. In Example 3-5, successive wafers are analyzed until a large
particle is detected. Then, X is the number of wafers analyzed. In the transmission of bits, X
might be the number of bits transmitted until an error occurs.

EXAMPLE 3-20 The probability that a bit transmitted through a digital transmission channel is received in
error is 0.1. Assume the transmissions are independent events, and let the random variable X
denote the number of bits transmitted until the first error.
Then, P(X 5) is the probability that the first four bits are transmitted correctly and the
fifth bit is in error. This event can be denoted as {OOOOE}, where O denotes an okay bit.
Because the trials are independent and the probability of a correct transmission is 0.9,

4
P1X 52 P1OOOOE2 0.9 0.1 0.066
Note that there is some probability that X will equal any integer value. Also, if the first trial is
a success, X 1. Therefore, the range of X is 51, 2, 3, p 6, that is, all positive integers.

Definition
In a series of Bernoulli trials (independent trials with constant probability p of a suc-
cess), let the random variable X denote the number of trials until the first success.
Then X is a geometric random variable with parameter 0 p 1 and

x 1
f 1x2 11 p2 p x 1, 2, p (3-9)

Examples of the probability mass functions for geometric random variables are shown in
Fig. 3-9. Note that the height of the line at x is (1 p) times the height of the line at x 1.
That is, the probabilities decrease in a geometric progression. The distribution acquires its
name from this result.