Discrete Random Variables¶

Overview¶

A random variable is a function that maps outcomes from a sample space to real numbers. A discrete random variable takes on a countable number of distinct values.

Definition¶

A random variable \(X\) is formally defined as a function:

\[ X : \Omega \longrightarrow \mathbb{R} \]

where \(\Omega\) is the sample space and \(\mathbb{R}\) is the set of real numbers.

A discrete random variable takes on a countable set of distinct values \(\{x_1, x_2, x_3, \ldots\}\). Examples include the result of rolling a die or the number of heads in a series of coin flips.

Distribution of a Discrete Random Variable¶

Imagine each outcome \(\omega \in \Omega\) as having a "brick" of a certain weight attached to it, representing the probability of that outcome. When we apply the random variable \(X\), we move the brick from \(\omega\) to the position \(X(\omega)\) on the real line.

After transferring all bricks, the arrangement of weights along \(\mathbb{R}\) defines the distribution of \(X\):

\[ \begin{aligned} \mathbb{P}(X = a) &= \text{Weight of the bricks at } a \\ \mathbb{P}(X \in A) &= \text{Weight of the bricks in the set } A \end{aligned} \]

Probability Mass Function (PMF)¶

For a discrete random variable \(X\), the PMF assigns a probability to each specific value:

\[ p_{x_i} = P(X = x_i) = \text{Weight of the brick at } x_i \]

The PMF must satisfy:

\(p_{x_i} \geq 0\) for all \(i\)
\(\sum_i p_{x_i} = 1\)

Examples¶

Example: PMF of a Fair Die¶

Let \(X\) represent the outcome of rolling a fair six-sided die:

\[ P(X = x) = \frac{1}{6}, \quad \text{for } x = 1, 2, 3, 4, 5, 6 \]

Example: Number of Heads in 3 Coin Flips¶

Let \(X\) represent the number of heads when flipping a fair coin 3 times. The possible values are \(\{0, 1, 2, 3\}\):

\[ \begin{aligned} P(X = 0) &= \frac{1}{8} \\ P(X = 1) &= \frac{3}{8} \\ P(X = 2) &= \frac{3}{8} \\ P(X = 3) &= \frac{1}{8} \end{aligned} \]

Example: Baseball Cards¶

Hugo plans to purchase packs of baseball cards until he obtains his favorite player's card. He can afford at most four packs, and each pack has a 0.2 probability of containing the card. Let \(X\) be the number of packs Hugo buys.

Solution:

\[ \begin{aligned} P(X=1) &= 0.2 \\ P(X=2) &= 0.8 \times 0.2 = 0.16 \\ P(X=3) &= 0.8^2 \times 0.2 = 0.128 \\ P(X=4) &= 1 - P(X=1) - P(X=2) - P(X=3) = 0.512 \end{aligned} \]

Therefore:

\[ \begin{aligned} P(X \geq 2) &= 1 - P(X=1) = 0.8 \\ P(X = 4) &= 0.512 \end{aligned} \]

Note that \(P(X=4) = 0.512\) includes both the probability of finding the card on the 4th pack and the probability of never finding it—Hugo stops at 4 regardless.

Example: Difference on Two 3-Sided Dice¶

Let \(D = |D_1 - D_2|\) where \(D_1, D_2\) are rolls of 3-sided dice. The nine equally likely outcomes yield:

\(D_1 \backslash D_2\)	1	2	3
1	0	1	2
2	1	0	1
3	2	1	0

\[ P(D=0) = \frac{3}{9}, \quad P(D=1) = \frac{4}{9}, \quad P(D=2) = \frac{2}{9} \]

Python Implementation¶

import numpy as np
import matplotlib.pyplot as plt

def plot_pmf(values, probabilities, title="PMF"):
    """Plot the probability mass function."""
    fig, ax = plt.subplots(figsize=(12, 3))
    ax.bar(values, probabilities, width=0.4, alpha=0.7, edgecolor='black')
    ax.set_xlabel('x')
    ax.set_ylabel('P(X = x)')
    ax.set_title(title)
    ax.spines['top'].set_visible(False)
    ax.spines['right'].set_visible(False)
    plt.tight_layout()
    plt.show()

# Fair die PMF
values = [1, 2, 3, 4, 5, 6]
probs = [1/6] * 6
plot_pmf(values, probs, "PMF of a Fair Die")

# Coin flip PMF (3 flips, counting heads)
from math import comb
n = 3
values = list(range(n + 1))
probs = [comb(n, k) * (0.5**k) * (0.5**(n-k)) for k in values]
plot_pmf(values, probs, "PMF: Number of Heads in 3 Coin Flips")

# Baseball cards PMF
values = [1, 2, 3, 4]
probs = [0.2, 0.16, 0.128, 0.512]
plot_pmf(values, probs, "PMF: Baseball Card Packs Purchased")

Key Takeaways¶

A discrete random variable maps outcomes to a countable set of real numbers.
The PMF gives the probability of each possible value and must sum to 1.
The "brick" metaphor provides intuition: each outcome carries a weight (probability), and the random variable relocates these weights to the real line.