03/12/2020

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

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:
**Random experiment**- daily mail;**Outcome**- the actual mail received;**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**.

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).