3.1.1 Two Types of Random Variables

• In Section 1.2, we distinguished between two types of data:
  • Data resulting from observations on a counting variable.
  • Data obtained by observing values of a measurement variable.
• A slightly more formal distinction characterizes two different types of random variables:
  • Counting variables (discrete random variables).
  • Measurement variables (continuous random variables).

discrete random variable

• A discrete random variable is defined as:
  • An rv whose possible values:
    • either constitute a finite set,
    • or can be listed in an infinite sequence (countably infinite) with a first element, a second element, and so on.
• A random variable is continuous if both of the following conditions apply:
  1. Its set of possible values consists of:
     • All numbers in a single interval on the number line (possibly infinite, e.g., from $-\infty$ to $\infty$), or
     • All numbers in a disjoint union of such intervals (e.g., $\left[0,10\right]\cup \left[20,30\right]$).
  2. No possible value of the variable has positive probability:
     • That is, $P(X=c)=0$ for any possible value $c$.

• Any interval on the number line contains:
  • An infinite number of numbers
  • No way to create an infinite listing of all these values
    • Reason: There are just too many of them
• Second condition describing a continuous random variable:
  • Counterintuitive implication:
    • Total probability of zero for all possible values
  • Clarification (to be discussed in Chapter 4):
    • Intervals of values have positive probability
    • Probability of an interval decreases to zero as:
    • Width of the interval shrinks to zero

EX 3.6

• Studying basic properties of discrete random variables (rv’s):
  • Requires:
  • Tools of discrete mathematics
    • Summation
    • Differences
• Studying continuous random variables:
  • Requires:
  • Continuous mathematics
    • Calculus
    • Integrals
    • Derivatives