Random Variables Definition

ABSTRACT

A Random Variable is a formal way to quantify outcomes. It is a function that maps every possible outcome in a sample space to a real number, allowing us to perform mathematical operations like averaging and summation on probabilistic events.

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Definition

A random variable $X$ is a function from the sample space $S$ to the set of real numbers $\mathbb{R}$:

$$X:S\to \mathbb{R}$$

The Distribution

The distribution of a random variable $X$ (specifically the Probability Mass Function for discrete variables) is the function that assigns a probability to each possible value $r$:

$$r\to P(X=r)$$

TIP

Think of the distribution as the probability that an outcome results in a specific random variable value.

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Expectation

The expectation ($E[X]$), also known as the average or expected value, is the center of mass of the distribution. It can be calculated in two ways:

1. Summing over outcomes:

   $$E(X)=\sum _{s\in S}P(s)X(s)$$

2. Summing over values (The Distribution Method):

   $$E(X)=\sum _{r\in X(S)}P(X=r)\cdot r$$

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Example: Sum of Two Dice

Let $X$ be the random variable representing the sum of the pips of 2 fair dice.

• Mapping:
  • $X(5,2)=7$
  • $X(3,3)=6$
• Distribution: The likelihood of each sum varies based on the number of ways to achieve it.
  • $P(X=7)=\frac{6}{36}=\frac{1}{6}$
  • $P(X=9)=\frac{4}{36}=\frac{1}{9}$

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Types of Random Variables

Type	Description	Example
Discrete	Values are countable (integers).	Number of heads in 10 flips.
Continuous	Values exist on a continuum (intervals).	The exact height of a person.
Indicator	Takes only values 0 or 1.	$X=1$ if a die is even, $X=0$ otherwise.

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Related Notes

• Distributions: High-level overview of how probability mass is spread.
• Expected Value: Deeper dive into the properties of $E[X]$.
• Linearity of Expectation: A method to find the expectation of a sum of random variables without knowing their joint distribution.
• Binomial Distribution: A specific, famous discrete random variable distribution.