03/10/2020

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

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**.**Afte**r 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.

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.