What Are Random Variables and How Do We Visualize Their Distributions?
How can we model the outcome of a random process?
What’s the difference between discrete and continuous probability?
And how do we visualize all of that?
This post answers these questions — with intuitive charts and real-world examples.
📚 This post is part of the "Intro to Statistics" series
🔙 Previously: Understanding Independence and Bayes’ Rule
🎲 What Is a Random Variable?
A random variable is a numerical outcome of a random phenomenon.
It can take different values depending on the situation — like the result of a die roll, the temperature in your city, or a person’s height.
🧱 Types of Random Variables
| Type | Description | Examples |
|---|---|---|
| Discrete | Takes a countable number of values | # of calls/day, die roll result |
| Continuous | Takes any value within an interval (infinite possibilities) | Height, temperature, weight |
📊 How Do We Work With Random Variables?
We use probability distributions to describe how likely each outcome is.
A probability distribution can be expressed as:
- A table
- A graph
- An equation
Depending on the variable type, we use:
| Type | Distribution Function |
|---|---|
| Discrete | Probability Mass Function (PMF) |
| Continuous | Probability Density Function (PDF) |
🔍 Visual: PMF (Discrete Distribution)
Each bar shows the probability of an exact outcome.
🔍 Visual: PDF (Continuous Distribution)
The area under the curve (not the height) represents probability.
You can’t directly say \( P(X = 5) \); it’s always \( P(a \le X \le b) \).
⚖️ Why Are Discrete Probabilities Simpler?
With discrete random variables, calculating probabilities is straightforward — you can just add up the values:
\( P(X = 2 \text{ or } X = 3) = P(X = 2) + P(X = 3) \)
In contrast, with continuous variables, you need to integrate the area under the curve — which often requires formulas or software.
📈 Cumulative Distribution Function (CDF)
The Cumulative Distribution Function answers:
What is the probability that \( X \) is less than or equal to some value?
We can compute CDFs for both discrete and continuous variables.
🧪 Example: CDF (Discrete)
| x | P(X = x) | P(X ≤ x) |
|---|---|---|
| 1 | 0.1 | 0.1 |
| 2 | 0.3 | 0.4 |
| 3 | 0.2 | 0.6 |
| 4 | 0.25 | 0.85 |
| 5 | 0.15 | 1.0 |
Each step adds the probability from the previous value.
📊 Example: CDF (Continuous)
This curve shows P(X ≤ x) for every point — and it always increases.
📉 Distribution vs Cumulative: Visual Comparison
| View | What It Shows |
|---|---|
| PDF / PMF | Probability of individual values (or areas) |
| CDF | Cumulative probability up to a certain point |
🎨 Visual Comparison
PDF → Use the area under curve to find probability
CDF → Read probability directly from the graph
📌 Key Properties of CDF
- Always increases (never decreases)
- Final value = 1
- You can find \( x \) for a given probability — or the other way around
🎯 What Is a Quantile?
A quantile tells us the value at a certain cumulative probability.
- The median is the 0.5 quantile → 50% of values lie below
- The 0.9 quantile means 90% of values are below that point
🔍 Visual Example
If the 90th percentile is 8.1, then \( P(X \le 8.1) = 0.90 \)
🧠 Level Up: Why CDFs Are Powerful
🧠 Try It Yourself: Random Variables & Distributions
Q1: What distinguishes a discrete random variable from a continuous one?
💡 Show Answer
Discrete variables take countable values; continuous variables can take infinitely many values within a range.
Q2: What is the name of the distribution function for discrete random variables?
💡 Show Answer
Probability Mass Function (PMF).
Q3: Why is it easier to compute probabilities with discrete variables?
💡 Show Answer
Because we can directly sum the individual probabilities without needing integration.
Q4: What does the CDF tell us?
💡 Show Answer
It gives the cumulative probability that a variable is less than or equal to a certain value.
Q5: What is a quantile?
💡 Show Answer
The value below which a certain proportion of the data falls — for example, the median is the 0.5 quantile.
✅ Summary
| Concept | Description |
|---|---|
| Random Variable | Represents numeric outcome of a random event |
| Discrete | Countable outcomes (use PMF) |
| Continuous | Infinite outcomes (use PDF) |
| PMF / PDF | Describe probability distribution |
| CDF | Accumulated probability up to x |
| Quantile | Inverse of CDF — get x for a given probability |
🔜 Up Next
In the next post, we’ll explore summary statistics like:
- Mean
- Variance
- Standard deviation
- Expected value
These help us describe how a probability distribution behaves.
Stay tuned!