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Chapter 3: Problem 4

Let \(X=\) the number of nonzero digits in a randomly selected zip code. What are the possible values of \(X\) ? Give three possible outcomes and their associated \(X\) values.

Short Answer

Expert verified
Possible values of \(X\) are 0, 1, 2, 3, 4, and 5.

Step by step solution

01

Understanding the Problem

A zip code is typically a 5-digit number. In this problem, we are asked to determine how many nonzero digits there can be in a randomly selected zip code. The value of \(X\) represents the number of these nonzero digits.
02

Identifying Possible Values of X

As a 5-digit zip code can have digits ranging from 0 to 9, we need to determine how many of the 5 digits can be nonzero. This means \(X\) can take any integer value from 0 (all digits are zeros, such as 00000) to 5 (all digits are non-zero, such as 12345). Therefore, the possible values of \(X\) are 0, 1, 2, 3, 4, and 5.
03

Providing Three Possible Outcomes

To exemplify, consider three different zip codes:1. **50000**: This zip code has one nonzero digit, so \(X = 1\).2. **40203**: This zip code has three nonzero digits, so \(X = 3\).3. **67890**: This zip code has five nonzero digits, so \(X = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Variable
A random variable is a way to quantify uncertain events. It assigns numerical values to outcomes of a random event or experiment. In the context of our exercise, the random variable, denoted by \(X\), represents the number of nonzero digits in a randomly chosen zip code.

Random variables can be
  • Discrete: These random variables can take on a countable number of distinct values, like whole numbers. \(X\) in our exercise is discrete, as it only assumes whole number values 鈥� the number of nonzero digits in zip codes.
  • Continuous: These random variables can take on an infinite number of values within a given range. They are not applicable in our zip code example.
When dealing with a discrete random variable like \(X\) in zip codes, it is crucial to identify the range of possible values, as outlined in the step-by-step solution. This understanding sets the foundation for further analysis, such as calculating probabilities.
Discrete Distribution
A discrete distribution describes how probabilities are assigned to each possible value of a discrete random variable. In our example, we want to determine how the number of nonzero digits, represented by random variable \(X\), distributes across different zip codes.

For discrete distributions:
  • The sum of the probabilities of all possible outcomes must equal 1.
  • Each individual probability value must be between 0 and 1.
For instance, if you analyze a large set of zip codes, you can observe a pattern of how \(X\) behaves and assign probabilities to each possible value from 0 to 5. A simple example: if 20% of sampled zip codes have three nonzero digits, \(P(X=3) = 0.2\).

Discrete distributions enable you to predict outcomes, assess risks, and make informed decisions based on patterns observed in data.
Problem Solving
Problem solving in probability involves breaking down a problem into understandable steps, analyzing possible outcomes, and determining the values that a random variable can assume. Our task concerning the number of nonzero digits in zip codes exemplifies such a process.

The three-step solution provides a structured approach:
  • Understanding the Problem: Recognize what is being asked 鈥� here, it's counting nonzero digits.
  • Identifying Possible Values: List all potential outcomes, like we did for \(X\) ranging from 0 to 5.
  • Providing Examples: Demonstrate scenarios with concrete examples, such as different zip codes highlighting various values of \(X\).
Approaching problems this way ensures clarity in thought and precision in problem-solving, enabling you to confidently tackle similar exercises in probability and other fields of mathematics.

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Most popular questions from this chapter

Suppose that \(30 \%\) of all students who have to buy a text for a particular course want a new copy (the successes!), whereas the other \(70 \%\) want a used copy. Consider randomly selecting 25 purchasers. a. What are the mean value and standard deviation of the number who want a new copy of the book? b. What is the probability that the number who want new copies is more than two standard deviations away from the mean value? c. The bookstore has 15 new copies and 15 used copies in stock. If 25 people come in one by one to purchase this text, what is the probability that all 25 will get the type of book they want from current stock? [Hint: Let \(X=\) the number who want a new copy. For what values of \(X\) will all 25 get what they want?] d. Suppose that new copies cost \(\$ 100\) and used copies cost \(\$ 70\). Assume the bookstore currently has 50 new copies and 50 used copies. What is the expected value of total revenue from the sale of the next 25 copies purchased? Be sure to indicate what rule of expected value you are using. [Hint: Let \(h(X)=\) the revenue when \(X\) of the 25 purchasers want new copies. Express this as a linear function.]

A personnel director interviewing 11 senior engineers for four job openings has scheduled six interviews for the first day and five for the second day of interviewing. Assume that the candidates are interviewed in random order. a. What is the probability that \(x\) of the top four candidates are interviewed on the first day? b. How many of the top four candidates can be expected to be interviewed on the first day?

A second-stage smog alert has been called in a certain area of Los Angeles County in which there are 50 industrial firms. An inspector will visit 10 randomly selected firms to check for violations of regulations. a. If 15 of the firms are actually violating at least one regulation, what is the pmf of the number of firms visited by the inspector that are in violation of at least one regulation? b. If there are 500 firms in the area, of which 150 are in violation, approximate the pmf of part (a) by a simpler pmf. c. For \(X=\) the number among the 10 visited that are in violation, compute \(E(X)\) and \(V(X)\) both for the exact pmf and the approximating pmf in part (b).

An individual who has automobile insurance from a certain company is randomly selected. Let \(Y\) be the number of moving violations for which the individual was cited during the last 3 years. The pmf of \(Y\) is \begin{tabular}{l|cccc} \(y\) & 0 & 1 & 2 & 3 \\ \hline\(p(y)\) & \(.60\) & \(.25\) & \(.10\) & \(.05\) \end{tabular} a. Compute \(E(Y)\). b. Suppose an individual with \(Y\) violations incurs a surcharge of \(\$ 100 Y^{2}\). Calculate the expected amount of the surcharge.

Suppose \(E(X)=5\) and \(E[X(X-1)]=27.5\). What is a. \(E\left(X^{2}\right)\) ? \(\left[\right.\) Hint: \(E[X(X-1)]=E\left[X^{2}-X\right]=\) \(\left.E\left(X^{2}\right)-E(X)\right] ?\) b. \(V(X)\) ? c. The general relationship among the quantities \(E(X)\), \(E[X(X-1)]\), and \(V(X) ?\)
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