# Probability Distributions, Mass Functions and Cumulative Distribution Functions

02/17/2021

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## Outline

• The following topics will be covered in this lecture:

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

## Random Variables 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 $$x$$

### Random variables continued 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.