In the previous article, we introduced the definitions of probability by Laplace and Kolmogorov.
Now we move on to probability distributions and probability density functions (PDFs).

Random Variables and Probability Distributions

First, we need the concept of a random variable.
A random variable $X$ is a way of assigning a numerical value to the outcome of a random experiment.

• Example: Roll a die → let $X$ be the number on the die ($X \in \{1,2,3,4,5,6\}$)
• Example: Drop a needle on the interval [0,1] → let $X$ be the position where it lands ($X \in [0,1]$)

A probability distribution describes which values $X$ can take and with what probability.
In other words, it’s like a “map” of the probabilities.

Discrete Case (Example: A Die)

If we roll a die once, the possible outcomes are $1,2,3,4,5,6$.
Defining $X$ as the die result, the probability distribution is:

As a table:

Outcome $k$	1	2	3	4	5	6
Probability $P(X=k)$	1/6	1/6	1/6	1/6	1/6	1/6

Such a discrete distribution is described by the Probability Mass Function (PMF).

Continuous Case (Example: Dropping a Needle on [0,1])

Now consider dropping a needle randomly on the line segment from 0 to 1 cm.
If $X$ is the position where the needle lands, what is $P(X=0.5)$?

Surprisingly, the probability of landing exactly at 0.5 cm is 0.
Instead, we consider probabilities over intervals:

Here, the probability is determined by a Probability Density Function (PDF).

For example, in the uniform distribution on [0,1], the density is:

And probabilities are calculated as areas under the curve:

So probability is the integral of density over an interval.

Discrete vs Continuous Distributions

Feature	Discrete Distribution (PMF)	Continuous Distribution (PDF)
Values handled	Individual values	Intervals
Probability of a single point	$P(X=x)$ is meaningful	$P(X=x)=0$
How to calculate	Summation	Integration

Summary

• A random variable $X$ represents outcomes of random experiments as numbers
• A probability distribution tells us the probabilities of $X$’s values
• PMFs describe discrete cases (dice, coins)
• PDFs describe continuous cases (intervals, measured by area)
• Probability is obtained as “the area under the density curve”

Probability Distributions and Densities

Interactive demonstration of discrete and continuous probability distributions

Discrete (Die Roll)Continuous (Uniform)Comparison

Discrete Distribution (Die Roll PMF)

PMF Property: Each individual outcome has a specific probability.
P(X = 3) = 1/6 ≈ 0.167

Key Concepts

Discrete Distribution (PMF)

• Each outcome has a specific probability
• Sum of all probabilities = 1
• P(X = specific value) ≠ 0

Continuous Distribution (PDF)

• Probability = area under the curve
• Total area under PDF = 1
• P(X = specific value) = 0

Mathematical Notes

Discrete: P(X = k) gives exact probability for outcome k

Continuous: P(a ≤ X ≤ b) = ∫_a^b f(x)dx

Key insight: For continuous distributions, probability is measured as area, not height!