More Discrete Random Variables and The Continuous Normal Distribution

Introduction

In the last unit you were introduced to two SPECIAL types of discrete random variables. The characteristic that makes them really special is the fact that the probability for each trial is the SAME. Both were an example from a larger family of distributions called DISCRETE random variables. Now we learn much more about the not-so-special discrete random variables. The probability will be changing for different values of the variable and all the information will be organized and stored in table form.

Of course, if there are variables containing "discrete" in the name, there must also be variables that are NOT discrete. The dictionary definition of discrete is "apart or detached from others, separate, or distinct." If that is true then the other type of variable must assume values that are "attached and not separate." The mathematical term for such a distribution is CONTINUOUS. We cannot use tables to summarize this type of variable because they take all values of integers, fractions and decimals, irrationals, and all real numbers. We need another model capable of suggesting infinite possible values that are connected. The most important example of a continuous random variable is the normal curve, which you may know as the "bell curve" since so many of our real life experiences can be modeled using this symmetric curve.

Essential Questions

What is a random variable?
What is a discrete random variable?
How are summary stats found for discrete distributions?
How do we combine stats for two or more discrete random variables?
What is a continuous random variable?
How are summary stats found for continuous distributions?
Have you ever heard of the bell curve?
What is a z-score and what do z-scores measure?
When does a data value become unusual?

Key Terms

The following key terms will help you understand the content in this module.

Continuous random variable- can take any numerical value within a range of values where the range can be infinite or bounded at either or both ends

Normal curve- bell-shaped, unimodal, symmetric shape depicting the normal distribution of 68-95-99.7

Normal distribution- most common distribution shape in statistics that serves as a standard of comparison for other distribution shapes

Inflection point- point along the normal curve where it intersects with the first +/- standard deviations

68-95-99.7 Empirical Rule- 68% of data is contained between +/- 1 standard deviation, 95%of data is contained between +/- 2 standard deviations, 99.7% of data is contained between +/- 3 standard deviations of mean

Percentile- the pth percentile of a distribution is the value such that p percent of the observations fall at or below it

IMAGES CREATED BY GAVS