Probability Distributions, Mass Functions and Cumulative Distribution Functions

02/17/2021

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Outline

  • The following topics will be covered in this lecture:

    • Random variables
    • Probability distributions
    • Probability mass functions
    • Cumulative distribution functions

Random Variables

Probability distribution for two coin flips with x number of heads.

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

  • Let us recall the idea of a random variable.
  • Prototypically, we can consider the coin flipping example from the motivation:
    • \( X \) is the number heads in two coin flips.
  • Every time we repeat two coin flips \( X \) can take a different value due to many possible factors:
    • how much force we apply in the flip;
    • air pressure;
    • wind speed;
    • etc…
  • The result is so sensitive to these factors that are beyond our ability to control, we consider the result to be by chance.
  • Before we flip the coin twice, the value of \( X \) has yet-to-be determined.
  • After we flip the coin twice, the value of \( X \) is fixed and possibly known.
  • Formally we will define:
  • Random Variable
    A random variable is a function that assigns a real number to each outcome in the sample space of a random experiment.
  • Notation
    A random variable is denoted by an uppercase letter such as \( X \). After an experiment is conducted, the measured value of the random variable is denoted by a lowercase letter such as \( x \)

Random variables continued

Random variables are the numerical measure of the outcome of a random process.

Courtesy of Ania Panorska CC

  • Suppose we are considering our sample space \( \mathbf{S} \) of all possible outcomes of a random process.
  • Then for any particular outcome of the process,
    • e.g., for the coin flips one outcome is \( \{H,H\} \),
  • mathematically the random variable \( X \) takes the outcome to the numerical value \( x=2 \) in the range \( \mathbf{R} \).
  • Note: \( X \) must always take a numerical value.
  • Because a random variable takes a numerical value (not categorical), we must consider the units that \( X \) takes:
    • Discrete random variable is a random variable with a finite (or countably infinite) range.
      • In particular, the unit of \( X \) cannot be arbitrarily sub-divided.

Random variables continued