Probability distributions part III

03/12/2020

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Outline

  • The following topics will be covered in this lecture:
    • Poisson distribution
    • Continuous random variables
    • Uniform distribution
    • Normal distributions

Poisson distribution motivation

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

Courtesy of Ania Panorska CC

  • Consider modeling the amount of mail you receive each day.
  • Suppose that expected number of pieces of mail on a given day will remain constant across any given day;
    • i.e., the actual ammount will have variation but the expected amount is always the same, e.g., 4 pieces.
  • Suppose as well that the arrival times of all pieces of mail are independent;
    • i.e., how many pieces of mail you receive one day does not affect future days.
  • If we wanted to model this kind of random experiment, we would call:
    1. Random experiment - daily mail;
    2. Outcome - the actual mail received;
    3. Random variable - total number of pieces of mail.
  • Let’s suppose that the standard deviation for the number of pieces of mail in a given day is also constant, equal to the square root of the expected number, \( \sqrt{4}=2 \).
  • If \( x \) is the random variable above, typically we would model its behavior in terms of a Poisson distribution.

Poisson distribution motivation continued

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

Courtesy of Ania Panorska CC

  • The example on the last slide of the daily mail is on particular example of a Poisson distributed varible because:
    • the random variable \( x \) measures the outcome of some random process over an interval;
      • i.e., we measured the ammount of mail over one day.
    • the outcome in any particular interval is considered to be by chance;
      • i.e., the actual mail we recieve on a given day is basically by chance.
    • The outcome in one interval is independent of the outcome in any other interval;
      • i.e., the mail recieved one day doesn’t affect the mail recieved any other day.
    • But in addition, the expected value for the random variable \( x \) is the same in any interval,
      • i.e., on every given day, we might expect to have \( 4 \) pieces of mail arriving (possibly more or less in reality).

Poisson distribution motivation continued