6.1 Introduction: Continuous Random Variables
Learning Objectives
By the end of this chapter, the student should be able to:
- Recognize and understand continuous probability density functions.
- Understand that continuous probability distributions are smooth curves or functions in which the area under the curve represents probability.
- Recognize and apply the uniform probability distribution.
- Recognize and apply the exponential probability distribution.
- Recognize and apply the normal probability distribution.
- Recognize and apply the standard normal probability distribution.
- Compare normal probabilities by converting to the standard normal distribution.
Future engineers and scientists may encounter examples such as the velocity of particles in fluid dynamics, the temperature distribution in a material under thermal analysis, or the electrical conductivity across a semiconductor. Additionally, they may explore the distribution of wind speeds in meteorology, the lifespan of electronic components, or the distance traveled by a vehicle between failures in reliability engineering. The field of reliability depends on a variety of continuous random variables. These examples demonstrate the diverse applications of continuous random variables, highlighting their significance in understanding and predicting real-world phenomena.
Random Variables – Counted or Measured?
The values of discrete and continuous random variables can be ambiguous. For example, if X is equal to the number of miles (to the nearest mile) you drive to work, then X is a discrete random variable. You count the miles. If X is the distance you drive to work, then you measure values of X and X is a continuous random variable. For a second example, if X is equal to the number of books in a backpack, then X is a discrete random variable. If X is the weight of a book, then X is a continuous random variable because weights are measured. How the random variable is defined is very important.
