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Chapter 3: Q4E (page 95)

Let X = the number of nonzero digits in a randomly selected 4-digit PIN that has no restriction on the digits. What are the possible values of X? Give three possible outcomes and their associated X values.

Short Answer

Expert verified

The possible values of X are:

\(X = \left\{ \begin{array}{l}0,\,\,\,{\rm{if all digits are zero}}\\1,\,\,\,\,{\rm{if three digits are zero}}\\2,\,\,\,{\rm{if two digits are zero}}\\3,\,\,\,{\rm{if one digit is zero}}\\4,\,{\rm{if}}\,{\rm{all digits are non - zero}}\end{array} \right.\)

The three possible outcomes and their associated X values are:

4-digit PIN

X

3123

4

4780

3

1000

1

Step by step solution

01

Given information

A random variable X is defined as the number of non-zero digits in a randomly selected PIN that has 4 numbers either zero or non-zero.

02

Determine the possible values of X

There are four places in a 4-digit PIN in which either zero or non-zero digits can be placed.

So, the minimum possible value of the random variable is 0 and the maximum possible value is 4.

The random variable X is defined by,

\(X = \left\{ \begin{array}{l}0,\,\,\,{\rm{if all digits are zero}}\\1,\,\,\,\,{\rm{if three digits are zero}}\\2,\,\,\,{\rm{if two digits are zero}}\\3,\,\,\,{\rm{if one digit is zero}}\\4,\,{\rm{if}}\,{\rm{all digits are non - zero}}\end{array} \right.\)

03

List three possible outcomes with their associated values of X

Suppose three 4-digit PIN are selected randomly, and then the possible outcomes are 3123, 4780 and 1000.

Following table represent the possible outcomes and their associated values of X which represent the number of non-zero digits in 4-digit PIN.

4-digit PIN

X

3123

4

4780

3

1000

1

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Most popular questions from this chapter

An instructor who taught two sections of engineering statistics last term, the first with\({\rm{20}}\)students and the second with\({\rm{30}}\), decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first\({\rm{15}}\)graded projects. a. What is the probability that exactly\({\rm{10}}\)of these are from the second section? b. What is the probability that at least\({\rm{10}}\)of these are from the second section? c. What is the probability that at least\({\rm{10}}\)of these are from the same section? d. What are the mean value and standard deviation of the number among these\({\rm{15}}\)that are from the second section? e. What are the mean value and standard deviation of the number of projects not among these first\({\rm{15}}\)that are from the second section?

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. What is the probability that

a. At most 6 of the calls involve a fax message?

b. Exactly 6 of the calls involve a fax message?

c. At least 6 of the calls involve a fax message?

d. More than 6 of the calls involve a fax message?

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.

Show that E(X) = np when X is a binomial random variable. (Hint: First express E(X) as a sum with lower limit x =\({\rm{1}}\). Then factor out np, let y = x\({\rm{ - 1}}\)so that the sum is from y = 0 to y = n\({\rm{ - 1}}\), and show that the sum equals\({\rm{1}}\).)

Of the people passing through an airport metal detector, .5% activate it; let \(X = \) the number among a randomly selected group of 500 who activate the detector.

a. What is the (approximate) pmf of X?

b. Compute \({\bf{P}}(X = 5)\)

c. Compute \({\bf{P}}(5 \le X)\)
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