1️⃣ What is Probability?

Probability is the mathematics of uncertainty. It tells us how likely an event is to occur — from flipping a coin to forecasting customer churn.

0≤P(E)≤10 \le P(E) \le 1

Where:

  • 0 means the event never happens,
  • 1 means it always happens,
  • Values in between show uncertainty.

🧩 Real-Life Examples

EventProbabilityMeaning
Tossing a coin → Heads0.550% chance
Rolling a die → 61/6~16.7% chance
Rain tomorrow0.330% chance of rain

💡 Why It Matters in Data Science

In data science, probability quantifies uncertainty — crucial for:

  • Predictive modeling (e.g., how likely a customer will buy)
  • Anomaly detection (how rare is this event?)
  • Model confidence and risk estimation

But plain probability has limits...

2️⃣ Why Probability Alone Has Limitations

In real life, most data isn’t simple or countable — it’s continuous.

Example:

  • Dice → 6 distinct outcomes ✅
  • Human height → 160.23 cm, 160.24 cm, 160.25 cm… → infinite possibilities ❌

So asking “What’s the probability someone’s height = 170.00 cm?” doesn’t make sense (it’s technically zero).

To solve this, we use probability distributions — functions that describe how probabilities are spread across possible values.

3️⃣ Probability Distribution Function (Overview)

A probability distribution describes how the values of a random variable are distributed — i.e., how likely each outcome is.

There are two types:

TypeFunctionVariable TypeExample
PMFProbability Mass FunctionDiscreteDice, number of customers
PDFProbability Density FunctionContinuousHeight, time, temperature

Both lead to a CDF (Cumulative Distribution Function), showing total probability up to a value.

4️⃣ Probability Mass Function (PMF)

📘 Definition

PMF gives the probability that a discrete random variable takes a specific value.

P(X=x)P(X = x)

🎲 Example: Rolling a Die

X (Outcome)P(X=x)
11/6
21/6
31/6
41/6
51/6
61/6

Each outcome has equal probability.

📊 PMF Graph (Discrete Probability)  

                        

👉 Caption: “The Probability Mass Function shows probability spikes at discrete points (1 to 6). Each bar height = P(X=x).”

💻 In Data Science

  • PMFs are used whenever you deal with countable or categorical data:
  • Binary classification: 0 (not spam) or 1 (spam) → Bernoulli distribution
  • Count events: number of transactions per hour → Poisson distribution
  • Text classification: word occurrences → Multinomial distribution

5️⃣ Probability Density Function (PDF)

📘 Definition

For continuous variables, we use a PDF, which describes how probability is distributed over a range.

P(a≤X≤b)=∫abf(x) dxP(a \le X \le b) = \int_a^b f(x)\,dx

The total area under the PDF curve = 1.

🧍 Example: Human Height Distribution

Heights (in cm) across adults follow a normal (bell-shaped) distribution:

  • Most values cluster around the mean (e.g., 170 cm)
  • Fewer people are much shorter or taller

📈 PDF Graph (Continuous Probability)


👉 Caption: “The area under the curve between two points gives the probability of falling in that range. The curve’s height indicates density, not direct probability.”

💻 In Data Science

PDFs appear in almost every continuous modeling problem:

Use CaseExample
Regression modelingResidual errors often follow a normal PDF
Gaussian Mixture Models (GMM)Each cluster represented by a PDF
Kernel Density Estimation (KDE)Smooth visualization of data distribution
Sensor noise modelingContinuous variation modeled via normal PDF

6️⃣ Cumulative Distribution Function (CDF)

📘 Definition

The CDF gives the total probability up to a value X:

F(x)=P(X≤x)F(x) = P(X \le x)

Derived from:

  • PMF: F(x)=∑i≤xP(X=i)F(x) = \sum_{i \le x} P(X=i)
  • PDF: F(x)=∫−∞xf(t)dtF(x) = \int_{-\infty}^x f(t)dt

🕒 Example: Delivery Times

Delivery Time (min)F(x) = P(X ≤ x)
100.10
200.35
300.70
400.90
501.00

F(30) = 0.70 → 70% of deliveries take 30 minutes or less.

📈 CDF Graph (Cumulative Probability)

👉 Caption: “The CDF starts at 0 and rises to 1, representing accumulated probability up to each point.”

💻 In Data Science

CDFs help in thresholding, percentile analysis, and decision-making:

Use CaseExample
Anomaly detectionDetect outliers beyond 95th percentile
Logistic regressionThe sigmoid curve = CDF of logistic distribution
Confidence intervalsDerived from cumulative probability
A/B testingCompare cumulative results over time

7️⃣ How to Read & Compare PMF, PDF, and CDF

Here’s what it really means 👇

We have three different “views” of probability, depending on the type of data and the question we ask.

ConceptVariable TypeGraph ShapeWhat It Really ShowsExample Interpretation
PMFDiscreteBars / SpikesProbability at exact values“The chance of rolling a 3 on a die is 1/6.”
PDFContinuousSmooth CurveProbability density across ranges“Most people’s height clusters around 170 cm.”
CDFBothRising Curve (0→1)Total probability up to a point“70% of deliveries take ≤ 30 minutes.”

Now, how to read them in practice 👇

  • PMF → “At this point, what’s the probability?”
  • PDF → “Between these two points, how likely is it?” (area under curve)
  • CDF → “Up to this point, how much probability has accumulated?”

📊 Think of it like this:

ConceptIntuitive Analogy
PMFLooking at individual bars in a histogram
PDFLooking at the smooth outline of that histogram
CDFWatching that histogram fill up from left to right

So in short:
👉 PMF = one point of probability
👉 PDF = range of probabilities
👉 CDF = total probability accumulated up to that range

8️⃣ Common Beginner Confusions

QuestionClarification
Why is the probability at a single point in PDF = 0?Because there are infinite possible values in continuous data.
Why does the CDF always increase?Probability accumulates — it can’t decrease.
How is PMF different from PDF?PMF = discrete values, PDF = continuous range.
Why is the area under PDF = 1?It represents total probability across all outcomes.

9️⃣ Real-World Data Science Use Cases

FunctionReal-World UseExample
PMFCounting discrete outcomesPredicting number of app downloads per hour
PDFModeling continuous variationEstimating sensor error in IoT systems
CDFDecision thresholdsSetting risk cutoffs in credit scoring

🔟 Mini Case Study — Predicting Food Delivery Times”

Let’s slow down and walk through what’s happening step-by-step so you see why we use PDF and CDF together.

🎯 Goal:

Predict how long food deliveries take — and tell users “Your food will likely arrive within X minutes.”

🧩 Step-by-Step Explanation

Step 1: Collect real delivery times (continuous data)

  • Gather thousands of delivery records.
  • Delivery time (in minutes) is a continuous variable (e.g., 23.5, 28.2, 31.8...).

Step 2: Fit a PDF (Normal Distribution)

  • When you plot all those delivery times as a histogram, it forms a bell-shaped curve (normal distribution).
  • This curve is the PDF (Probability Density Function) — it shows where delivery times are most likely.
    • Peak (center) = most common delivery time
    • Left and right tails = very fast or very late deliveries

✅ Example: The peak of the curve might be around 30 minutes, meaning that’s the average delivery time.

Step 3: Derive the CDF (Cumulative Distribution Function)

  • The CDF takes that same PDF curve and accumulates probability from left to right.

  • At any point x, it tells you:

    F(x)=P(delivery time≤x)F(x) = P(\text{delivery time} \le x)

    So F(30) means “the probability the delivery takes 30 minutes or less.”

Step 4: Calculate F(30)
If F(30) = 0.70 → that means 70% of all deliveries take 30 minutes or less.

✅ That’s your service performance metric — the “70th percentile” of delivery time.

Step 5: Use CDF for ETA predictions
You can now use this information to make customer-facing statements:

  • “There’s a 70% chance your food arrives within 30 minutes.”
  • “There’s an 85% chance it arrives within 35 minutes.”

💡 The Insight (Why it’s powerful):

Using PDFs + CDFs together helps you move from averages to probabilistic predictions.

Instead of saying:

“Delivery time = 30 minutes.”

You can say:

“Your delivery will likely arrive within 30 minutes (70% confidence) or within 35 minutes (85% confidence).”

That’s exactly how Uber Eats, DoorDash, and Swiggy calculate their live ETAs!

🔍 In Data Science Terms

  • PDF → Models the distribution of delivery times.
  • CDF → Converts that distribution into percentile-based probabilities.
  • CDF thresholds (like 70%, 90%) help define performance benchmarks and SLAs (service-level agreements).

🔍 FAQs

Q1. Can I use PMF for continuous data?
No — PMF is only for discrete data. Use PDF for continuous variables.

Q2. Why is PDF not the same as probability?
Because it’s a density. Only the area under the curve gives probability.

Q3. What’s the relationship between PDF and CDF?
CDF is the integral (sum) of PDF up to that point.

Q4. Where is CDF used in machine learning?
In logistic regression, calibration plots, ROC curves, and anomaly detection.

Q5. Can CDF ever go beyond 1?
No. It always ranges from 0 → 1 (total certainty).

🧩 Summary Table

FeaturePMFPDFCDF
TypeDiscreteContinuousBoth
RepresentsP(X=x)Density f(x)P(X≤x)
GraphBarsCurveRising curve
TotalΣ = 1∫ = 1Ends at 1
Used ForCountable eventsContinuous modelingThresholds, percentiles

🏁 Conclusion

Understanding PMF, PDF, and CDF is the key to mastering probability in data science.
They allow you to model uncertainty, analyze risk, and interpret data distributions — the foundation of intelligent, probabilistic systems.

In short:
PMF tells you what’s likely,
PDF shows how it spreads,
CDF explains how it accumulates.