Discrete Random Variables

Probability Theory

When Outcomes Are Numbers, Not Just Names

A discrete random variable takes on a countable number of distinct values, each with an associated probability. It transforms qualitative outcomes into quantitative analysis.

PMF — The probability mass function assigns a probability to each possible value
CDF — The cumulative distribution function gives P(X ≤ x) for any value
Expected value — The probability-weighted average of all possible outcomes
Variance — How much the random variable fluctuates around its expected value

Discrete random variables are the gateway from probability theory to statistical inference.

What is a Discrete Random Variable?

Definition

A discrete random variable takes on a countable number of distinct values, each with an associated probability. The probabilities of these values are described by the probability mass function (PMF).

Probability Mass Function (PMF)

P(X = x_i) = p_i \quad \text{where} \quad \sum_{i} p_i = 1 \quad \text{and} \quad p_i \geq 0

Here,

$X$ =The discrete random variable
$x_i$ =The i-th possible value
$p_i$ =Probability that X = x_i

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# Example: fair die
values = np.array([1, 2, 3, 4, 5, 6])
probs = np.array([1/6, 1/6, 1/6, 1/6, 1/6, 1/6])

print("PMF of a Fair Die:")
for v, p in zip(values, probs):
    print(f"  P(X={v}) = {p:.4f}")
print(f"  Sum = {probs.sum():.4f}")

Cumulative Distribution Function (CDF)

CDF of Discrete RV

F(x) = P(X \leq x) = \sum_{x_i \leq x} p_i

Here,

$F(x)$ =Cumulative probability up to x
$p_i$ =PMF at value x_i

# CDF of a fair die
cumulative_probs = np.cumsum(probs)
print("\nCDF of a Fair Die:")
for v, cp in zip(values, cumulative_probs):
    print(f"  F({v}) = P(X ≤ {v}) = {cp:.4f}")

Expected Value and Variance

Expected Value

E[X] = \mu = \sum_{i} x_i \cdot p_i

Here,

$E[X]$ =Expected value (mean) of X
$x_i$ =Possible values
$p_i$ =Associated probabilities

Variance

\text{Var}(X) = \sigma^2 = \sum_{i} (x_i - \mu)^2 \cdot p_i

Here,

$\text{Var}(X)$ =Variance of X
$\mu$ =Expected value of X
$\sigma$ =Standard deviation

expected_value = np.sum(values * probs)
variance = np.sum((values - expected_value)**2 * probs)
std_dev = np.sqrt(variance)

print(f"\nE[X] = {expected_value:.4f}")
print(f"Var(X) = {variance:.4f}")
print(f"SD(X) = {std_dev:.4f}")

Common Discrete Distributions

fig, axes = plt.subplots(2, 2, figsize=(10, 8))

# Bernoulli
x_ber = np.array([0, 1])
p_ber = [0.3, 0.7]
axes[0, 0].bar(x_ber, p_ber, color='steelblue')
axes[0, 0].set_title('Bernoulli(p=0.7)')
axes[0, 0].set_xlabel('x')
axes[0, 0].set_ylabel('P(X=x)')

# Binomial
x_bin = np.arange(0, 11)
p_bin = stats.binom.pmf(x_bin, n=10, p=0.5)
axes[0, 1].bar(x_bin, p_bin, color='steelblue')
axes[0, 1].set_title('Binomial(n=10, p=0.5)')
axes[0, 1].set_xlabel('x')

# Poisson
x_poi = np.arange(0, 12)
p_poi = stats.poisson.pmf(x_poi, mu=3)
axes[1, 0].bar(x_poi, p_poi, color='steelblue')
axes[1, 0].set_title('Poisson(λ=3)')
axes[1, 0].set_xlabel('x')

# Geometric
x_geo = np.arange(1, 11)
p_geo = stats.geom.pmf(x_geo, p=0.3)
axes[1, 1].bar(x_geo, p_geo, color='steelblue')
axes[1, 1].set_title('Geometric(p=0.3)')
axes[1, 1].set_xlabel('x')

plt.tight_layout()
plt.savefig('discrete-distributions.png', dpi=150)
plt.show()

Distribution	PMF	Mean	Variance
Bernoulli(p)	$p^x(1-p)^{1-x}$	$p$	$p(1-p)$
Binomial(n,p)	$\binom{n}{x}p^x(1-p)^{n-x}$	$np$	$np(1-p)$
Poisson(λ)	$\frac{e^{-\lambda}\lambda^x}{x!}$	$\lambda$	$\lambda$
Geometric(p)	$(1-p)^{x-1}p$	$1/p$	$(1-p)/p^2$