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Chapter 3: Q6E (page 98)

Starting at a fixed time, each car entering an intersectionis observed to see whether it turns left (L), right (R), orgoes straight ahead (A). The experiment terminates assoon as a car is observed to turn left. Let X = the numberof cars observed. What are possible X values? List five outcomes and their associated X values.

Short Answer

Expert verified

The possible X values are 1,2,3,4, and so on i.e.\(X \in \left\{ {1,2,3,4, \ldots } \right\}\)

Sample space

X

L

1

AL

2

RAL

3

AARL

4

AARRL

5

Step by step solution

01

Given information

The random variable X is defined as the number of cars observed entering an intersection.

The experiment terminates as soon as a car turns left is been observed.

02

Determine the possible X values

The random variable X can only take on positive values:\(1,2,3,4, \ldots \)as X counts the number of cars observed.

Therefore, the random variable is defined by:

\(X = \left\{ {1,2,3,4, \ldots } \right\}\)

The random variable X takes infinite set of values in this experiment.

03

List five possible outcomes with their associated values of X.

Here, X is defined as the number of cars observed and it terminated when the car turns left.

The samplespace of the experimentis\(S = \left\{ {L,AL,RAL,AARL,AARRL} \right\}\).

Following table, represent the possible outcomes and their associated values of X.

Sample space

X

L

1

AL

2

RAL

3

AARL

4

AARRL

5

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Most popular questions from this chapter

Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives circuit boards in batches of five. Two boards are selected from each batch for inspection. We can represent possible outcomes

of the selection process by pairs. For example, the pair (1, 2) represents the selection of boards 1 and 2 for inspection.

a. List the ten different possible outcomes.

b. Suppose that boards 1 and 2 are the only defective boards in a batch. Two boards are to be chosen at random. Define X to be the number of defective boards observed among those inspected. Find the probability distribution of X.

c. Let F(x) denote the cdf of X. First determine \(F\left( 0 \right) = P\left( {X \le 0} \right)\), F(1), and F(2); then obtain F(x) for allother x.

Eighteen individuals are scheduled to take a driving test at a particular DMV office on a certain day, eight of whom will be taking the test for the first time. Suppose that six of these individuals are randomly assigned to a particular examiner, and let X be the number among the six who are taking the test for the first time. a. What kind of a distribution does X have (name and values of all parameters)? b. Compute P(X\({\rm{ = 2}}\)), P(X\( \le {\rm{ 2}}\)), and P(X\( \ge {\rm{ 2}}\)). c. Calculate the mean value and standard deviation of X.

Twenty pairs of individuals playing in a bridge tournament have been seeded \({\rm{1,}}....{\rm{,20}}\). In the first part of the tournament, the \({\rm{20}}\) are randomly divided into 10 east鈥� west pairs and \({\rm{10}}\) north鈥搒outh pairs. a. What is the probability that x of the top \({\rm{10}}\) pairs end up playing east鈥搘est? b. What is the probability that all of the top five pairs end up playing the same direction? c. If there are \({\rm{2}}\)n pairs, what is the pmf of X = the number among the top n pairs who end up playing east鈥搘est? What are E(X) and V(X)?

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table.

y

45

46

47

48

49

50

51

52

53

54

55

p(y)

.05

.10

.12

.14

.25

.17

.06

.05

.03

.02

.01

a. What is the probability that the flight will accommodateall ticketed passengers who show up?

b. What is the probability that not all ticketed passengerswho show up can be accommodated?

c. If you are the first person on the standby list (whichmeans you will be the first one to get on the plane ifthere are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y, and state which Y value is associated with each one.
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