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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

Consider a collection \({{\rm{A}}_{\rm{1}}}{\rm{,}}...{\rm{,}}{{\rm{A}}_k}\) of mutually exclusive and exhaustive events, and a random variable \({\rm{X}}\) whose distribution depends on which of the \({{\rm{A}}_{\rm{i}}}{\rm{'s}}\) occurs (e.g., a commuter might select one of three possible routes from home to work, with \({\rm{X}}\) representing the commute time). Let \({\rm{E(X|}}{{\rm{A}}_{\rm{i}}}{\rm{)}}\) denote the expected value of \({\rm{X}}\) given that the event \({{\rm{A}}_{\rm{i}}}\) occurs. Then it can be shown that \({\rm{E(X) = }}\sum {{\rm{E(X|}}{{\rm{A}}_{\rm{i}}}{\rm{)}} \cdot {\rm{P(}}{{\rm{A}}_{\rm{i}}}{\rm{)}}} \), the weighted average of the individual 鈥渃onditional expectations鈥� where the weights are the probabilities of the partitioning events.

a. The expected duration of a voice call to a particular telephone number is \({\rm{3}}\) minutes, whereas the expected duration of a data call to that same number is \({\rm{1}}\) minute. If \({\rm{75\% }}\) of all calls are voice calls, what is the expected duration of the next call?

b. A deli sells three different types of chocolate chip cookies. The number of chocolate chips in a type \({\rm{i}}\) cookie has a Poisson distribution with parameter \({{\rm{\mu }}_{\rm{i}}}{\rm{ = i + 1 (i = 1,2,3)}}\). If \({\rm{20\% }}\) of all customers purchasing a chocolate chip cookie select the first type, \({\rm{50\% }}\) choose the second type, and the remaining \({\rm{30\% }}\) opt for the third type, what is the expected number of chips in a cookie purchased by the next customer?

After shuffling a deck of \({\rm{52}}\) cards, a dealer deals out\({\rm{5}}\). Let \({\rm{X = }}\) the number of suits represented in the five-card hand.

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(Hint: \({\rm{p(1) = 4P}}\) (all are spades), \({\rm{p(2) = 6P}}\) (only spades and hearts with at least one of each suit), and \({\rm{p(4)}}\) \({\rm{ = 4P(2}}\) spades \({\rm{{C}}}\) one of each other suit).)

b. Compute\({\rm{\mu ,}}{{\rm{\sigma }}^{\rm{2}}}\), and\({\rm{\sigma }}\).

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate\(\alpha = 8\)per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter\(\mu = 8t\). a. What is the probability that exactly 6 small aircraft arrive during a one-hour period? At least\(6\)? At least\({\bf{10}}\)? b. What are the expected value and standard deviations of the number of small aircraft that arrive during a\({\bf{90}}\)-min period? c. What is the probability that at least\({\bf{20}}\)small aircraft arrive during a\({\bf{2}}.{\bf{5}}\)-hour period? That at most\({\bf{10}}\)arrive during this period?

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a. Does\({\rm{X}}\)have a binomial distribution? Explain your reasoning.

b. What is the probability that exactly three of the ten test results are positive?
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