Probability distributions part II

03/10/2020

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Outline

  • The following topics will be covered in this lecture:
    • Review of random variables
    • Review of distributions
    • Binomial distribution
    • Parameters of the binomial distribution

Random variables

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

Courtesy of Ania Panorska CC

  • 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 – is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.

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 – these take numerical values that are in counting units.
      • In particular, the unit of \( x \) cannot be arbitrarily sub-divided.
        • We can think of “how many coin flips heads” is measured in counting units because \( 1.45 \) heads does not make sense.
      • However, the values \( x \) takes don’t strictly need to be whole numbers;
        • the units just cannot be arbitrarily sub-divided.
      • The scale of units for \( x \) can be finite or infinite depending on the problem.

Random variables continued