Continuous Random Variables¶

Overview¶

A continuous random variable can take on any value within a continuous range (an interval or union of intervals on the real line). Unlike discrete random variables, the probability of any single specific value is zero—instead, probabilities are defined over intervals.

Definition¶

A continuous random variable \(X\) can take on infinitely many possible values within a given range. Examples include heights, weights, temperatures, and waiting times.

For continuous random variables, the weight (probability) is spread continuously along the real line rather than concentrated at specific points.

Probability Density Function (PDF)¶

For a continuous random variable \(X\), the PDF \(f(x)\) describes the density of probability at each point:

\[ f(x)\,dx = \text{Weight of the bricks within the interval } [x, x + dx] \]

Key properties of the PDF:

\(f(x) \geq 0\) for all \(x\)
\(\int_{-\infty}^{\infty} f(x)\,dx = 1\)
\(P(a \leq X \leq b) = \int_a^b f(x)\,dx\)

Note that \(f(x)\) itself is not a probability—it can exceed 1. Only the area under the curve gives probabilities.

Key Difference from Discrete Variables¶

For a discrete random variable, we can ask \(P(X = a)\) and get a positive answer. For a continuous random variable:

\[ P(X = a) = 0 \quad \text{for any specific value } a \]

This is because there are infinitely many possible values, and the "weight" at any single point is zero. We can only meaningfully ask about the probability that \(X\) falls within a range.

Example: Normal PDF¶

The most important continuous distribution is the normal distribution with mean \(\mu\) and variance \(\sigma^2\). Its PDF is:

\[ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right) \]

The area under this curve between any two values gives the probability that \(X\) falls within that range.

Example: Amelia's Maximum Average Wait Time¶

The distribution of average wait times at drive-through restaurants is approximately normal with mean \(\mu = 185\) seconds and standard deviation \(\sigma = 11\) seconds. Amelia only uses restaurants in the bottom 10% of wait times. What is her maximum acceptable wait time?

Solution:

We need the 10th percentile (PPF at 0.1):

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

mu = 185
sigma = 11

max_wait = stats.norm(loc=mu, scale=sigma).ppf(0.1)
print(f"Maximum average wait time: {max_wait:.2f} seconds")

# Visualization
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 200)
pdf = stats.norm(loc=mu, scale=sigma).pdf(x)

fig, ax = plt.subplots(figsize=(12, 3))
ax.plot(x, pdf)
x_fill = np.linspace(mu - 3*sigma, max_wait, 100)
ax.fill_between(x_fill, stats.norm(loc=mu, scale=sigma).pdf(x_fill),
                alpha=0.3, color='r', label=f'Bottom 10% (≤ {max_wait:.1f}s)')
ax.spines[['right', 'top']].set_visible(False)
ax.spines['bottom'].set_position('zero')
ax.legend()
plt.show()

Comparing Discrete and Continuous Distributions¶

Feature	Discrete	Continuous
Values	Countable set	Uncountable (interval)
Probability at a point	\(P(X = a) > 0\) possible	\(P(X = a) = 0\) always
Probability function	PMF: \(p_{x_i}\)	PDF: \(f(x)\)
Probability of a range	\(\sum_{x_i \in [a,b]} p_{x_i}\)	\(\int_a^b f(x)\,dx\)
Total probability	\(\sum_i p_{x_i} = 1\)	\(\int_{-\infty}^{\infty} f(x)\,dx = 1\)

Key Takeaways¶

Continuous random variables take values in an interval; the probability of any single point is zero.
The PDF describes the "density" of probability—areas under the PDF curve give probabilities.
The PDF can exceed 1 at specific points, but the total area under the curve is always 1.
The normal distribution is the most widely used continuous distribution.