Kekkei - Advancing Financial Science For Everyone

Introduction

Random variables transform outcomes into numbers, enabling us to apply mathematical analysis to probability. They're the bridge between abstract sample spaces and practical calculations.

Learning Objectives:

Understand random variables as functions
Work with probability mass functions (PMFs)
Calculate probabilities using cumulative distribution functions (CDFs)
Implement random variables in Python

Random Variables

A random variable is a function $X: \Omega \rightarrow \mathbb{R}$ that maps each outcome in the sample space to a real number.

Notation: Uppercase letters ( $X$ , $Y$ ) for random variables, lowercase ( $x$ , $y$ ) for specific values.

Types:

Discrete: Takes countable values (integers, finite set)
Continuous: Takes values in an interval

Example: Die Roll

Sample space: $\Omega = \{⚀, ⚁, ⚂, ⚃, ⚄, ⚅\}$

Define $X$ = "number showing":

$X(⚀) = 1$
$X(⚁) = 2$
...
$X(⚅) = 6$

Now we can ask: $P(X = 4)$ , $P(X > 3)$ , $E[X]$ , etc.

Probability Mass Function

For discrete random variable $X$ , the PMF is:

$p_X(x) = P(X = x)$

Properties:

$p_X(x) \geq 0$ for all $x$
$\sum_{x} p_X(x) = 1$ (sum over all possible values)

python

import numpy as np
import matplotlib.pyplot as plt

# Die roll: PMF
values = [1, 2, 3, 4, 5, 6]
pmf = [1/6] * 6

print("PMF for fair die:")
for x, p in zip(values, pmf):
    print(f"  P(X = {x}) = {p:.4f}")

print(f"\nSum of PMF: {sum(pmf):.4f} (must equal 1)")

# Verify with simulation
rolls = np.random.randint(1, 7, size=10000)
for x in values:
    empirical_prob = np.mean(rolls == x)
    theoretical_prob = 1/6
    print(f"X={x}: empirical={empirical_prob:.4f}, theoretical={theoretical_prob:.4f}")

Cumulative Distribution Function

The CDF gives the probability that $X$ is at most $x$ :

$F_X(x) = P(X \leq x) = \sum_{t \leq x} p_X(t)$

Properties:

$F_X$ is non-decreasing
$\lim_{x \to -\infty} F_X(x) = 0$
$\lim_{x \to \infty} F_X(x) = 1$
$P(a < X \leq b) = F_X(b) - F_X(a)$

python

import numpy as np

# Die roll CDF
def die_cdf(x):
    """CDF for fair 6-sided die"""
    if x < 1:
        return 0
    elif x >= 6:
        return 1
    else:
        return int(np.floor(x)) / 6

# Compute CDF at various points
test_points = [0, 1.5, 3, 3.7, 6, 10]
print("CDF values:")
for x in test_points:
    cdf_val = die_cdf(x)
    print(f"  F_X({x}) = P(X ≤ {x}) = {cdf_val:.4f}")

# Calculate interval probability
a, b = 2, 5
prob_interval = die_cdf(b) - die_cdf(a-1)  # P(2 ≤ X ≤ 5)
print(f"\nP({a} ≤ X ≤ {b}) = F_X({b}) - F_X({a-1}) = {prob_interval:.4f}")

Key Takeaways

Random variable: Function from outcomes to numbers
PMF: $p_X(x) = P(X = x)$ for discrete RVs
CDF: $F_X(x) = P(X \leq x)$ , works for any RV
PMF sums to 1: $\sum_x p_X(x) = 1$
CDF properties: Non-decreasing, limits to 0 and 1

Next Lesson: Common discrete distributions (Bernoulli, Binomial, Poisson)!

Concept	Discrete Formula	Example
PMF	$p_X(x) = P(X=x)$	Die: $p_X(3) = 1/6$
CDF	$F_X(x) = \sum_{t \leq x} p_X(t)$	Die: $F_X(3) = 1/2$
Interval	$P(a < X \leq b) = F_X(b) - F_X(a)$	$P(2 < X \leq 5) = F_X(5) - F_X(2)$

IntroductionFocusStart Focus Mode

Random VariablesFocusStart Focus Mode

Example: Die Roll

Probability Mass FunctionFocusStart Focus Mode

Cumulative Distribution FunctionFocusStart Focus Mode

Key TakeawaysFocusStart Focus Mode

Introduction

Random Variables

Probability Mass Function

Cumulative Distribution Function

Key Takeaways