02/17/2021

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The following topics will be covered in this lecture:

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

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 \( x \)

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.