Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- PART ONE PROBABILITY IN ACTION
- PART TWO ESSENTIALS OF PROBABILITY
- 7 Foundations of probability theory
- 8 Conditional probability and Bayes
- 9 Basic rules for discrete random variables
- 10 Continuous random variables
- 11 Jointly distributed random variables
- 12 Multivariate normal distribution
- 13 Conditioning by random variables
- 14 Generating functions
- 15 Discrete-time Markov chains
- 16 Continuous-time Markov chains
- Appendix: Counting methods and ex
- Recommended reading
- Answers to odd-numbered problems
- Bibliography
- Index
10 - Continuous random variables
Published online by Cambridge University Press: 05 August 2012
- Frontmatter
- Contents
- Preface
- Introduction
- PART ONE PROBABILITY IN ACTION
- PART TWO ESSENTIALS OF PROBABILITY
- 7 Foundations of probability theory
- 8 Conditional probability and Bayes
- 9 Basic rules for discrete random variables
- 10 Continuous random variables
- 11 Jointly distributed random variables
- 12 Multivariate normal distribution
- 13 Conditioning by random variables
- 14 Generating functions
- 15 Discrete-time Markov chains
- 16 Continuous-time Markov chains
- Appendix: Counting methods and ex
- Recommended reading
- Answers to odd-numbered problems
- Bibliography
- Index
Summary
In many practical applications of probability, physical situations are better described by random variables that can take on a continuum of possible values rather than a discrete number of values. Examples are the decay time of a radioactive particle, the time until the occurrence of the next earthquake in a certain region, the lifetime of a battery, the annual rainfall in London, and so on. These examples make clear what the fundamental difference is between discrete random variables taking on a discrete number of values and continuous random variables taking on a continuum of values. Whereas a discrete random variable associates positive probabilities to its individual values, any individual value has probability zero for a continuous random variable. It is only meaningful to speak of the probability of a continuous random variable taking on a value in some interval. Taking the lifetime of a battery as an example, it will be intuitively clear that the probability of this lifetime taking on a specific value becomes zero when a finer and finer unit of time is used. If you can measure the heights of people with infinite precision, the height of a randomly chosen person is a continuous random variable. In reality, heights cannot be measured with infinite precision, but the mathematical analysis of the distribution of heights of people is greatly simplified when using a mathematical model in which the height of a randomly chosen person is modeled as a continuous random variable.
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- Information
- Understanding Probability , pp. 318 - 359Publisher: Cambridge University PressPrint publication year: 2012