From bars to curves

In the discrete world, every outcome has its own probability. Roll a die: P(3) = 1/6. Count heads: P(X = 5) is a number you can write down. But what if outcomes are real numbers — like measuring the exact time a bus arrives, or the precise length of a phone call?

What do you think?

You pick a completely random real number between 0 and 1. What is the probability of picking exactly 0.5?

In the continuous world, the probability of any exact value is zero. Instead, probabilities live in intervals.

Probability density functions

For discrete random variables, we listed probabilities with a PMF. For continuous random variables, we describe how probability is "smeared" across the number line using a density function.

Probability Density Function (PDF)

A continuous random variable $X$ has a probability density function $f(x)$ satisfying:

$f(x) \geq 0$ for all $x$
$\int_{-\infty}^{\infty} f(x)\, dx = 1$
For any interval $[a, b]$ : $P(a \leq X \leq b) = \Large\int_a^b f(x)\, dx$

The PDF is not a probability itself — it's a density. The height of the curve tells you how concentrated probability is near that value. Actual probability only comes from the area under the curve over an interval.

PDF Explorer

Left bound (a): 3.0

010

Right bound (b): 7.0

010

P(a < X < b)

0.8176

Interval

[3.0, 7.0]

Drag the interval endpoints around. The shaded area is the probability.

Discrete vs. continuous: a side-by-side

Feature	Discrete	Continuous
Outcomes	Countable (integers, categories)	Uncountable (real numbers)
Distribution	PMF: $P(X = k)$	PDF: $f(x)$
Probabilities from	Summing: $\sum P(X = k)$	Integrating: $\int f(x)\, dx$
P(X = exact value)	Can be positive	Always 0
Total	$\sum_k P(X=k) = 1$	$\int_{-\infty}^{\infty} f(x)\, dx = 1$

Because P(X = a) = 0 for continuous RVs, it doesn't matter whether intervals are open or closed: $P(a \lt X \lt b) = P(a \leq X \leq b)$ .

The cumulative distribution function

Just like in the discrete case, the CDF accumulates probability from the left:

CDF (Continuous)

$F(x) = P(X \leq x) = \Large\int_{-\infty}^x f(t)\, dt$ The PDF and CDF are related by differentiation: $f(x) = F'(x)$ wherever $F$ is differentiable.

Properties of the CDF (same as in the discrete case):

$F$ is non-decreasing
$F(x) \to 0$ as $x \to -\infty$
$F(x) \to 1$ as $x \to +\infty$

If f(x) = 2x on [0,1] and 0 elsewhere, what is P(0 ≤ X ≤ 0.5)? (decimal, e.g. 0.42)

For the same PDF, what is P(X > 0.5)? (decimal, e.g. 0.42)

Explore the connection between PDF and CDF — drag a point and watch area accumulate, or compute interval probabilities:

PDF ↔ CDF Explorer

Distribution

Mode

x = 0.50: 0.50

-3.53.5

PDF: f(x) — shaded area = probability

CDF: F(x) = P(X ≤ x) — non-decreasing from 0 to 1

f(x)

0.3521

F(x) = P(X ≤ x)

0.6915

Slope F'(x) = f(x)

0.3521

Is it discrete or continuous?

Discrete or Continuous?

Can you list all possible values? If yes, it's discrete. If the values form a continuous range, it's continuous.

Question 1 of 6 | Score: 0/0

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Expectation and variance

The formulas from the discrete case carry over — just replace sums with integrals:

Continuous Expectation & Variance

$E[X] = \Large\int_{-\infty}^{\infty} x\, f(x)\, dx$ $\text{Var}(X) = E[X^2] - (E[X])^2 = \Large\int_{-\infty}^{\infty} (x - \mu)^2 f(x)\, dx$

All the rules you learned still hold:

Linearity: $E[aX + b] = aE[X] + b$
Variance scaling: $\text{Var}(aX + b) = a^2 \text{Var}(X)$
Independence: If $X \perp Y$ , then $E[XY] = E[X]E[Y]$ and $\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y)$

Summary

Concept	Key Idea
PDF	Density function; height ≠ probability
Probability	Area under the curve over an interval
P(X = exact value)	Always 0 for continuous RVs
CDF	$F(x) = \int_{-\infty}^x f(t)\, dt$
Expectation	Replace $\sum$ with $\int$

Mental shift: In the continuous world, stop asking "what's the probability of this value" and start asking "what's the probability of this region."

What's next

Now that we know what continuous distributions look like, let's meet the simplest one: the Uniform distribution — where every interval of the same length is equally likely.