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Chapter 6: Problem 1

What is a random variable?

Short Answer

Expert verified
A random variable represents values based on the outcomes of a random event.

Step by step solution

01

- Definition

A random variable is a variable whose values depend on outcomes of a random phenomenon. It is a fundamental concept in probability theory.
02

- Types

There are two main types of random variables: discrete and continuous. Discrete random variables have specific and countable values, while continuous random variables can take any value within a range.
03

- Examples

For a discrete random variable example, consider the roll of a fair six-sided die. The possible values are 1, 2, 3, 4, 5, and 6. For a continuous random variable, consider the time it takes for a computer to load a program, which can be any value within a certain range.

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Key Concepts

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

Probability Theory
To understand random variables, you first need to grasp the basic idea of probability theory. Probability theory is a branch of mathematics focused on analyzing random events. It鈥檚 about predicting the likelihood of various outcomes.
This theory helps us make sense of uncertain situations and compute probabilities. For example, if you toss a coin, there is a 50% chance (or probability) it will land heads up.
Probability theory is indispensable in many fields, from science and engineering to finance and everyday decision-making. It includes important concepts like random variables, probability distributions, and events.
All of these concepts work together to help describe and understand randomness.
Discrete Random Variables
A discrete random variable is one that has a countable number of possible values. Each value is distinct and separate from the others. Examples can help clarify this.
Imagine rolling a fair six-sided die. The random variable representing the outcome can only be one of the specific numbers: 1, 2, 3, 4, 5, or 6. These values are countable and finite.
Characteristics of discrete random variables include:
  • Specific, distinct values
  • A finite or countably infinite set of outcomes
  • Occurrences like rolling dice or flipping coins
Understanding discrete random variables is crucial for analyzing situations where outcomes are limited and separated by gaps.
Continuous Random Variables
Unlike discrete random variables, continuous random variables can take any value within a range. These values are not countable or distinct points but rather any number in a continuum. For example, consider the time it takes for a computer to load a program.
This time could be 2.5 seconds, 2.55 seconds, or any value between them, without any gaps.
Key features of continuous random variables are:
  • A continuous range of possible values
  • A need to use intervals rather than distinct points when measuring
  • Examples like weight, height, or temperature
Continuous random variables are essential in many real-world scenarios where measurements aren't restricted to whole numbers.

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Most popular questions from this chapter

(a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section \(6.1 ;\) (c) compute the mean and standard deviation, using the methods of this section; and \((d)\) draw a graph of the probability distribution and comment on its shape. $$ n=8, p=0.5 $$

Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. Three cards are selected from a standard 52 -card deck with replacement. The number of kings selected is recorded.

(a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section \(6.1 ;\) (c) compute the mean and standard deviation, using the methods of this section; and \((d)\) draw a graph of the probability distribution and comment on its shape. $$ n=10, p=0.2 $$

The number of hits to a website follows a Poisson process; hits occur at the rate of 1.4 per minute between 7:00 p.m. and 9:00 p.m. Compute the probability that the number of hits between 7:30 p.m. and 7:35 p.m. is (a) exactly seven. Interpret the result. (b) fewer than seven. Interpret the result. (c) at least seven. Interpret the result.

In 1898 , Ladislaus von Bortkiewicz published The Law of Small Numbers, in which he demonstrated the power of the Poisson probability law. Before his publication, the law was used exclusively to approximate binomial probabilities. He demonstrated the law's power, using the number of Prussian cavalry soldiers who were kicked to death by their horses. The Prussian army monitored 10 cavalry corps for 20 years and recorded the number \(X\) of annual fatalities because of horse kicks for the 200 observations. The following table shows the data: $$\begin{array}{ll}\hline \text { Number of } & \text { Number of Times } \boldsymbol{x} \\\\\text { Deaths, } \boldsymbol{x} & \text { Deaths Were Observed } \\\\\hline 0 & 109 \\\\\hline 1 & 65 \\\\\hline 2 & 22 \\\\\hline 3 & 3 \\\\\hline 4 & 1 \\\\\hline\end{array}$$ (a) Compute the proportion of years in which there were 0 deaths, 1 death, 2 deaths, 3 deaths, and 4 deaths. (b) From the data in the table, what was the mean number of deaths per year? (c) Use the mean number of deaths per year found in part (b) and the Poisson probability law to determine the theoretical proportion of years that 0 deaths should occur. Repeat this for \(1,2,3,\) and 4 deaths. (d) Compare the observed proportions to the theoretical proportions. Do you think the data can be modeled by the Poisson probability law?
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