CDF (Cumulative Distribution Function)

From Events to Random Variables

Many questions about a random number ask whether it stays below some level.

Probability is first defined for events, but many questions involve numerical values rather than yes-or-no outcomes. A random variable connects these two views by assigning a number to each outcome of an experiment.

Once a random variable is defined, statements about its value naturally form events. Expressions like

X \le x

,

X > x

, or

a < X \le b

describe collections of outcomes and therefore have probabilities attached to them.

question about a die roll X	as an event
at most 4?	X ≤ 4
more than 2?	X > 2
between 3 and 5?	2 < X ≤ 5

Answering each such question from scratch is tedious. One list that answered every "at most

x

" at once would answer the others too.

For a die, a running total of the probabilities builds that list.

A fair die shows each face with probability

\frac{1}{6}

. Go up the faces and keep a running total: the probability of rolling at most

x

.

x	P(X = x)	P(X ≤ x): running total
1	1/6	1/6
2	1/6	2/6
3	1/6	3/6
4	1/6	4/6
5	1/6	5/6
6	1/6	6/6 = 1

"More than

2

" is then

1 - \frac{2}{6} = \frac{4}{6}

, and "between

3

and

5

" is

\frac{5}{6} - \frac{2}{6} = \frac{3}{6}

— read off the same column.

Nothing in the running total was chosen: each entry is the last one plus one more probability.

So the column can only stay the same or grow, it starts at

0

before the smallest value and ends at

1

after the largest. Between faces nothing is added, so the total stays flat and then jumps at each face — a staircase.

The running total for a die, drawn: flat between faces, a jump of

\frac{1}{6}

at each face, ending at

1

. The same state is on the discrete CDF visualizer.

The definition

The cumulative distribution function of a random variable

X

gives, for each real number

x

, the probability that

X

is at most

x

:

F(x) = P(X \le x)

.

For the die,

F(4) = \frac{4}{6}

. The next two sections state the meaning and the formal definition in full.

A CDF is a function, defined for every real number, not only for the values X can take.

Its input is any number

x

and its output is a probability between

0

and

1

. For the die,

F(4.7) = \frac{4}{6}

and

F(-3) = 0

, even though the die never shows

4.7

or

-3

.

The cumulative distribution function is built exactly on this connection. It takes events of the form

X \le x

and assigns to each value

x

the probability of that event, linking random variables back to the event-based foundation of probability.

Running totals generalize; the jumps were particular to a die, and they make ≤ and < differ.

What generalizes: every random variable has a CDF, built the same way. What was particular: a die's values are separate, so its CDF climbs in jumps; a continuous variable such as a measured height has a CDF that rises smoothly. And at a jump, "at most" and "less than" are different questions.

question	value
P(X ≤ 4)	4/6
P(X < 4)	3/6: the jump at 4 is left out
P(X ≤ 4.7)	4/6: nothing new between 4 and 5

The rest of the page states the CDF's core meaning and formal definition, its notation and properties, the CDF of discrete, continuous and mixed variables, and how to use it to find probabilities.

CDF Defined (Core Meaning)

The cumulative distribution function describes how probability is distributed along the values of a random variable. For each possible value, it tells us how much probability has accumulated up to that point.

Instead of asking whether the random variable takes one specific value, the CDF answers a broader question: how likely it is that the value does not exceed a given level. As the value increases, the accumulated probability can only stay the same or grow.

This perspective is what makes the CDF so powerful. It focuses on accumulation rather than individual outcomes, providing a single, consistent way to describe the behavior of a random variable across its entire range.

Mathematical Definition

Formally, the cumulative distribution function of a random variable

X

is defined by

CDF Definition

F_X(x) = P(X \le x)

Learn more about this formula: CDF Definition →

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For each real number

x

, the CDF assigns the probability that the random variable takes a value less than or equal to

x

. The function

F_X

maps numbers on the real line to values between 0 and 1.

The choice of "less than or equal to" is not arbitrary. It ensures that the CDF behaves consistently across discrete, continuous, and mixed random variables, allowing a single definition to cover all cases.

CDF Notation

Notation

CDF Notation

One capital letter, one subscript, and one inequality sign that was chosen rather than inherited. The case of the letter is the whole distinction between a probability and a density, and every mark here is catalogued among the probability symbols.

X

,

x

,

P(\cdot)

and the density

f(x)

come from the probability function; the event braces

\{X \le x\}

that

P

measures from random variable notation.

F_X(x)

capital F sub X of x — the CDF of X at x

Capital

F

names the cumulative function, and the subscript names which variable it belongs to:

F_X(x) = P(X \le x)

, per Mathematical Definition above. The subscript earns its place the moment two variables share a page —

F_X

and

F_Y

are different functions.

CasesThe subscript is dropped when only one variable is in play, leaving plain

F(x)

; the argument is always a number on the real line, so

F

is defined everywhere even where the variable never lands.

Do not confuseThe density

f(x)

. One letter, two cases, two different objects:

F

accumulates and is bounded by

1

, while lowercase

f

is a rate that may exceed

1

— CDF vs PMF vs PDF below sets them side by side, and calculus ties them together as

F' = f

.

P(X \le x)

the probability that X is at most x

The inequality is a convention, not an accident: writing

\le

rather than

<

includes the point itself, which is what makes the CDF right-continuous and lets one definition serve discrete, continuous and mixed variables alike.

CasesFor continuous variables the choice is invisible —

P(X = x) = 0

, so

\le

and

<

agree; for discrete ones it decides whether a jump belongs to the step before or after it, which is exactly where Common Mistakes below finds its errors.

Do not confuseA strict inequality.

F(3)

includes the outcome

3

— the strict version

P(X < 3)

is what excludes it, and the two differ by exactly the mass sitting at the point.

F(b) - F(a)

F of b minus F of a

The idiom for interval probabilities: accumulating to

b

and subtracting what accumulated to

a

leaves

P(a < X \le b)

— the notation of Using the CDF to Compute Probabilities below, and the reason the CDF is a computational tool rather than a description.

CasesThe same shape as the evaluation bracket of definite integrals — and not by coincidence: for continuous variables

F(b) - F(a)

is exactly

\int_a^b f(x)\,dx

.

Do not confuseWhich endpoint is included. The difference gives

P(a < X \le b)

— open on the left, closed on the right; for discrete variables shifting to

P(a \le X \le b)

requires adding the mass at

a

back in.

All probability symbols →↑ Probability

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Notation · Cheat sheet

Random Variable Notation at a Glance

The marks from the notation cards across the random-variable pages: how each is read and what it means. Every group links to the full card on its page.

Random variables

X : \Omega \to \mathbb{R}

X maps outcomes to numbers

A random variable is a function, not a variable.

\{X \le x\}

the event that X is at most x

Braces turn a condition into an event.

X \sim \text{Bin}(n, p)

X is distributed as

Names the distribution and its parameters.

Probability functions

X

vs

x

the variable; a value

Capital for the random variable, lowercase for a number it takes.

P(\cdot)

the probability of

Takes an event, returns a number in

[0, 1]

.

p(x)

vs

f(x)

mass; density

p

is a chance (discrete);

f

is a rate (continuous).

PMF

\{x : p_X(x) > 0\}

the support of X

Every value that carries probability.

\sum_{x} p_X(x) = 1

the masses add to one

The sum runs over the support, not from a start to an end.

PDF

f_X(x)

the density of X at x

A density, not a probability; it can exceed

1

.

P(a \le X \le b)

X between a and b

The integral

\int_a^b f_X(x)\,dx

.

P(X = a) = 0

a single point

For a continuous variable, always zero.

CDFThis page

F_X(x)

the CDF of X at x

Equals

P(X \le x)

: everything up to and including

x

.

F(b) - F(a)

F of b minus F of a

Equals

P(a < X \le b)

.

Indicators

I_A, \; \mathbf{1}_A

the indicator of A

1

if

A

happens,

0

if not.

\mathbf{1}\{X \le x\}

one if the condition holds

The brace form takes a statement instead of an event.

N = \sum I_{A_i}

a count of events

How many of the

A_i

happened.

Same marks, different job

P

vs

p

Capital

P

takes an event; lowercase

p

is a mass function of a value.

F

vs

f

Capital

F

accumulates (the CDF); lowercase

f

is the density.

\sim

Is distributed as; in linear algebra, row-equivalent or similar.

\{X \le x\}

An event, not a set of numbers.

Random variable symbols table Probability

Random Variable Notation at a Glance·/probability/random-variablesLearn Math Class

Key Properties of the CDF

The cumulative distribution function obeys several fundamental properties that follow directly from the probability axioms.

Values between 0 and 1
For every $x$ , the CDF satisfies $0 \le F_X(x) \le 1$ .

Non-decreasing
As $x$ increases, $F_X(x)$ can only stay the same or increase. Accumulated probability never decreases.

Right-continuous
The value of the CDF at a point includes the probability at that point, which ensures consistent behavior for jumps in discrete cases.

Limits at infinity
As $x \to -\infty$ , $F_X(x) \to 0$ .
As $x \to +\infty$ , $F_X(x) \to 1$ .

Stated formally:

CDF Limits at Infinity

\lim_{x \to -\infty} F_X(x) = 0, \qquad \lim_{x \to +\infty} F_X(x) = 1

Learn more about this formula: CDF Limits at Infinity →

CDF Monotonicity

x_1 \leq x_2 \implies F_X(x_1) \leq F_X(x_2)

Learn more about this formula: CDF Monotonicity →

CDF Right-continuity

\lim_{y \to x^+} F_X(y) = F_X(x)

Learn more about this formula: CDF Right-continuity →

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Pitfall: Forgetting where the CDF is zero

Writing

F_X(x) = 1 - e^{-x}

for every real

x

gives

F_X(-1) = 1 - e \approx -1.72

, a negative probability, and

F_X(x) \to -\infty

as

x \to -\infty

. As a CDF it must read

F_X(x) = 0

for

x < 0

and

1 - e^{-x}

for

x \ge 0

.

A CDF given by a formula is usually written for the values the variable can actually take, such as

x \ge 0

for a waiting time, and that is where most of the work happens. The properties above, though, are statements about every real number

x

, including the values far to the left and far to the right of anything the variable can produce.

These properties are what distinguish a valid CDF from an arbitrary function and guarantee that it represents a genuine probability distribution. The table below names each property, states it formally, and gives the structural guarantee it provides.

Property	Statement	What it ensures
Bounded	0 ≤ F_X(x) ≤ 1 for every x	every value is a valid probability
Non-decreasing	F_X(x) stays the same or increases as x grows	accumulated probability never decreases
Right-continuous	the value at x includes the probability at x	jumps in discrete and mixed cases behave consistently
Limits at infinity	F_X(x) → 0 as x → −∞; F_X(x) → 1 as x → +∞	the function spans the full probability scale

These four properties are the whole test: every function that passes them is the CDF of some random variable.

Discrete Random Variables

For a discrete random variable, probability is concentrated at specific values. The CDF reflects this by increasing only at those values and remaining flat everywhere else.

In this case, the CDF is a step function. Each jump corresponds to a value the random variable can take, and the size of the jump equals the probability assigned to that value.

Formally, the CDF adds up the probabilities of all values up to

x

:

CDF from PMF

F_X(x) = \sum_{k \leq x} p_X(k)

Learn more about this formula: CDF from PMF →

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Pitfall: Reading the CDF as a point probability

Taking

F_X(3) = \frac{1}{2}

for a fair die as the chance of rolling a

3

counts the faces

1

,

2

and

3

together. The chance of a

3

alone is the jump at

3

:

F_X(3) - F_X(2) = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}

.

This makes the connection to the probability mass function clear: the PMF determines the jump sizes, while the CDF accumulates those jumps as the value increases. Reading the CDF from left to right shows how probability builds up one possible value at a time.

On a graph of the staircase, two different heights can be read at every possible value: the level of the step that starts there, and the size of the riser that leads up to it. The level is a running total collected from the far left; the riser belongs to the single value where it stands.

Binomial CDF, n = 10, p = 0.5: a staircase

The cumulative probability is flat between the possible values and jumps at each one; the height of the jump at k is exactly P(X = k), so the whole probability mass function can be read off the risers. Nothing accumulates between integers because a discrete variable puts no probability there. Switch distributions and compare the staircases on the discrete CDF explorer.

Continuous variables give a very different picture.

Continuous Random Variables

For a continuous random variable, probability is spread smoothly over intervals rather than concentrated at individual points. As a result, the CDF increases continuously instead of jumping.

In this setting, the CDF describes how probability accumulates as the value grows along the real line. The slope of this accumulation is captured by the probability density function, which indicates how rapidly probability is building at each point.

In symbols, the CDF accumulates the density, and the density is the slope of the CDF:

CDF from PDF

F_X(x) = \int_{-\infty}^{x} f_X(t)\,dt

Learn more about this formula: CDF from PDF →

PDF as CDF Derivative

f_X(x) = \frac{d}{dx} F_X(x)

Learn more about this formula: PDF as CDF Derivative →

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COROLLARY

A key consequence is that a continuous random variable assigns zero probability to any single exact value.

Pitfall: Reading the density as a probability

For

X

uniform on

[0, \frac{1}{2}]

the density is

f_X(x) = 2

, and taking

f_X(\frac{1}{4}) = 2

as

P(X \le \frac{1}{4})

gives a probability above

1

. The CDF gives it:

F_X(\frac{1}{4}) = 2 \cdot \frac{1}{4} = \frac{1}{2}

.

Probabilities are obtained only by looking at intervals, and the CDF provides a direct way to compute them.

Two functions appear in this section, and both take a number

x

and return a number. The CDF

F_X(x)

returns accumulated probability, so its values always lie between

0

and

1

. The density

f_X(x)

returns the rate at which that probability is building near

x

; where a lot of probability is packed into a short stretch, the rate is high. A narrow interval can therefore carry a large density and still hold very little probability.

Normal CDF, mean 0, standard deviation 1

Here the cumulative probability rises without any jumps: probability is spread over intervals, so P(X = x) is zero for every single point and the curve climbs smoothly from 0 toward 1. The steepest part of the S-curve sits where the density is largest. Move along the curve and read the accumulated probability on the continuous CDF visualizer.

Mixed distributions combine both behaviours.

Mixed Distributions

Some random variables do not fit neatly into purely discrete or purely continuous categories. They may assign positive probability to certain specific values while also spreading probability continuously over intervals.

Fact: A CDF with no jumps and no density

Write

X

in base

3

, each digit a fair choice of

0

or

2

: every single value has probability

0

, so the CDF never jumps, yet

X

stays in the Cantor set, of total length

0

, so there is no density. This CDF, the devil's staircase, climbs from

0

to

1

while staying flat on gaps of total length

1

inside

[0, 1]

.

The CDF handles this naturally. It combines flat regions, smooth increases, and sudden jumps within a single function. Jumps represent discrete probability masses, while smooth segments represent continuously distributed probability.

This is one reason the CDF is more general than the PMF or PDF. Even when neither of those fully describes a distribution on its own, the CDF still provides a complete and consistent representation.

A CDF that jumps and ramps

A random variable that equals 0 with probability 0.3 and otherwise spreads the remaining 0.7 evenly over (0, 2]. Its CDF is 0 to the left of 0, jumps by 0.3 at x = 0 (the filled dot: the value at 0 includes the jump), rises in a straight line from 0.3 to 1 across (0, 2], and stays at 1 after that. No PMF and no PDF describes this variable on its own, but one CDF does.

The jump gives P(X = 0) = 0.3; any interval inside (0, 2] gets 0.35 per unit of length.

Using the CDF to Compute Probabilities

One of the main advantages of the cumulative distribution function is that it allows probabilities to be computed directly from differences of values.

For any two numbers

a < b

, the probability that the random variable lies between them is obtained by subtracting accumulated probabilities:

Probability of Interval via CDF

P(a < X \le b) = F_X(b) - F_X(a)

Learn more about this formula: Probability of Interval via CDF →

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This works uniformly for discrete, continuous, and mixed random variables. One-sided probabilities are handled just as easily by reading the CDF at a single point.

Pitfall: Dropping the mass at the left endpoint

For a fair die, writing

P(2 \le X \le 4) = F_X(4) - F_X(2) = \frac{4}{6} - \frac{2}{6} = \frac{1}{3}

leaves out the face

2

. Subtracting

F_X(1)

keeps it:

\frac{4}{6} - \frac{1}{6} = \frac{1}{2}

.

For a discrete variable, the interval in the formula has to match the interval in the question exactly. The left side is open at

a

and closed at

b

, and the difference on the right reproduces precisely that: the mass sitting at

b

is counted, the mass sitting at

a

is not. Questions are phrased with every combination of

<

and

\le

, so each one is matched to this shape before any value of

F_X

is read.

For a continuous random variable the same probability is the area under the density:

Probability of Interval Continuous

P(a \leq X \leq b) = \int_{a}^{b} f_X(x)\,dx

Learn more about this formula: Probability of Interval Continuous →

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Because of this, the CDF often simplifies probability calculations by replacing sums or integrals with simple evaluations of a single function.

Continuous uniform CDF, a = 0, b = 10: a straight ramp

For a uniform variable the CDF is a straight line from 0 at a to 1 at b, so the probability of any interval inside [a, b] is the difference of the two ramp heights, which is the length of the interval divided by b minus a. The difference F(b) minus F(a) is the same calculation on every distribution, only the curve changes. Pick two points and read the difference on the continuous CDF visualizer.

The comparison with the PMF and PDF makes this role precise.

CDF vs PMF vs PDF

The CDF, PMF, and PDF describe probability distributions from different perspectives, but they are not interchangeable.

The CDF tracks accumulated probability. It tells how much probability lies at or below a given value and always exists for any random variable.

The PMF applies only to discrete random variables. It assigns probabilities to individual values, and the CDF is obtained by summing these probabilities up to a point.

The PDF applies only to continuous random variables. It describes how densely probability is spread, and the CDF is obtained by accumulating this density over an interval.

Because the CDF works in all cases, it serves as the most general representation of a probability distribution, with the PMF and PDF appearing as special cases derived from it. The table below sets the three side by side across what each function applies to, what it represents, and how it relates to the CDF.

Function	Applies to	What it represents	Relation to CDF
CDF — F_X(x)	any random variable	accumulated probability up to x	the function itself
PMF — p_X(x)	discrete variables only	probability at each individual value	CDF is the cumulative sum of PMF values
PDF — f_X(x)	continuous variables only	probability density at each point	CDF is the integral of PDF from −∞ to x

Visual Interpretation

The cumulative distribution function can be understood visually as probability accumulating along the number line.

Starting from the far left, the CDF begins at zero and increases as more possible values are included. In discrete cases, this accumulation appears as upward jumps. In continuous cases, it appears as a smooth rising curve. Mixed distributions show both behaviors together.

Reading the CDF from left to right shows how probability builds up. At any point, the height of the curve represents how much of the total probability lies at or below that value.

Common Mistakes

The cumulative distribution function is often misunderstood because it looks simple but encodes a lot of structure.

A common mistake is confusing the CDF with a density or mass function. The CDF does not describe how much probability sits at a point, but how much has accumulated up to that point.

Another frequent error is forgetting that the CDF is cumulative. Interpreting its value as a probability of an exact outcome leads to incorrect conclusions, especially in continuous cases.

In discrete distributions, jumps in the CDF are sometimes misread as arbitrary features. Each jump has a precise meaning: its size equals the probability assigned to that value.

Finally, it is easy to forget that every valid CDF must satisfy basic properties such as monotonicity and proper limits. Violating these properties means the function cannot represent a probability distribution. The table below pairs each common mistake with the correction that prevents it.

Mistake	Correction
Confusing CDF with a density or mass function	F_X(x) is probability accumulated up to x, not the probability at x
Forgetting the CDF is cumulative	reading F_X(x) as a point probability misreads it, especially in continuous cases where P(X = x) = 0
Misreading jumps in discrete CDFs	each jump has a precise meaning — its size equals the probability assigned to that value
Ignoring required properties	a function that is not monotonic or lacks the correct limits cannot represent a probability distribution

Two further mistakes come up once the CDF is used for calculations. Each one is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Reading the density as a probability

For a continuous variable, the density

f_X(x)

is evaluated at a point and its value is reported as a probability, either as

P(X \le x)

or as the chance of landing at

x

.

Wrong working: for

X

uniform on

[0, \frac{1}{2}]

, the density is

f_X(x) = 2

on that interval, so

P\left(X \le \frac{1}{4}\right) = f_X\left(\frac{1}{4}\right) = 2

Right working: the probability is the area under the density up to

\frac{1}{4}

, which is the value of the CDF there.

F_X\left(\frac{1}{4}\right) = \int_0^{1/4} 2\,dt = \frac{1}{2}

Why it happens: for a discrete variable the PMF value at a point is a probability, and the habit carries over to the density, which looks like the same kind of function and is written with almost the same letter as

F_X

.

How to check: a probability can never exceed

1

, while a density can. Here

f_X(x) = 2

at every point of

[0, \frac{1}{2}]

, although the whole interval holds probability

F_X(\frac{1}{2}) = 1

.

Dropping the mass at the left endpoint

For a discrete variable, the probability of a closed interval

a \le X \le b

is computed as

F_X(b) - F_X(a)

, the formula for the half-open interval

a < X \le b

.

Wrong working: for a fair die,

P(2 \le X \le 4) = F_X(4) - F_X(2) = \frac{4}{6} - \frac{2}{6} = \frac{1}{3}

Right working:

F_X(2)

already contains the face

2

, so subtracting it removes that face from the answer. Subtract the CDF one value lower instead.

P(2 \le X \le 4) = F_X(4) - F_X(1) = \frac{4}{6} - \frac{1}{6} = \frac{1}{2}

Why it happens: the formula

P(a < X \le b) = F_X(b) - F_X(a)

is remembered without its strict inequality on the left. For continuous variables the slip costs nothing, since

P(X = a) = 0

, so it goes unnoticed until a discrete problem.

How to check: count the values directly. The faces

2

,

3

and

4

give

\frac{3}{6} = \frac{1}{2}

.

Why the CDF Matters

THEOREM

The cumulative distribution function fully characterizes the distribution of a random variable. Knowing the CDF is enough to recover all probability statements about the variable.

It provides a single framework that works for discrete, continuous, and mixed distributions. This makes it a central object in probability theory and a natural bridge between different types of models.

Fact: Every continuous variable hides a uniform one

Feed a continuous

X

into its own CDF and

F_X(X)

is uniform on

[0, 1]

:

F_X(X) \le u

means

X

has not passed the point where

F_X

reaches

u

, which has probability

u

. Run backwards,

F_X^{-1}(U)

for a uniform

U

has CDF

F_X

, a standard way to simulate any distribution.

Many important concepts rely directly on the CDF, including quantiles, medians, percentiles, and probability intervals. In both theory and applications, the CDF is often the most convenient way to reason about probabilities.

A quantile reads the CDF backwards. The median is the value

m

with

F_X(m) = \frac{1}{2}

, and the

90

th percentile is the value where

F_X

first reaches

0.9

. For a continuous variable every level

u

between

0

and

1

is reached somewhere, so the CDF and its inverse pass freely between values of

X

and probabilities.

Connections to Other Probability Concepts

The CDF ties together several core ideas in probability.

• Events appear as statements of the form

X \le x

.
• Random variables provide the numerical structure the CDF describes.
• PMF and PDF are specific ways probability is distributed, both derived from the CDF.
• Probability axioms guarantee the basic properties of the CDF.
• Quantiles and percentiles are defined by inverting the CDF.
• Expectation and variance can be expressed using the CDF.

Through these connections, the CDF serves as a unifying object that links foundational probability concepts with practical calculations.

CDF Across the Three Cases

The cumulative distribution function applies uniformly to discrete, continuous, and mixed random variables, yet its visual and structural behavior differs in each setting. The table below collects these three cases — naming the CDF's shape, how it changes as x grows, and the companion function (PMF or PDF) that captures the local detail behind each pattern.

Variable type	CDF shape	How it changes	Companion function
Discrete	step function — flat between values, jumps at possible values	increases only at the possible values of X	PMF — jump size at x equals p_X(x)
Continuous	smooth, continuously increasing curve, no jumps	increases continuously across the real line	PDF — slope of the CDF equals f_X(x)
Mixed	combines flat regions, smooth segments, and isolated jumps	jumps at discrete masses and smooth growth over continuous parts	partial PMF (at the masses) and partial PDF (over the continuous parts)

CDF FAQ

Why is the CDF defined with ≤ rather than <?

+

It is a deliberate choice, not an accident. Including the point itself is what makes the CDF right-continuous, and it lets a single definition cover discrete, continuous and mixed variables alike. For continuous variables the choice is invisible, since P(X = x) = 0. For discrete ones it decides whether a jump belongs to the step before it or the step after.Read more →

Does F(b) − F(a) include the endpoint a?

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No. The difference gives P(a < X ≤ b) — open on the left, closed on the right. For a continuous variable that distinction costs nothing, since single points carry no probability. For a discrete variable it matters: to get P(a ≤ X ≤ b) you must add back the mass sitting at a, which F(a) has already subtracted away.Read more →

What is the difference between F(x) and f(x)?

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One letter, two cases, two different objects. Capital F accumulates probability and is bounded by 1; lowercase f is a rate and may exceed 1. Calculus ties them together — f is the derivative of F, and F is the integral of f — but they answer different questions: F gives a probability directly, f does not.Read more →

What is the survival function?

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The complement of the CDF: P(X > x) = 1 − F(x). Where the CDF accumulates probability up to a point, the survival function measures what is left beyond it. It leaves out the point itself, so it differs from P(X ≥ x) by exactly the mass sitting at x — invisible for continuous variables, but a real gap at a discrete jump. It is standard in reliability and time-to-event work.Read more →

Is the CDF defined for values the variable can never take?

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Yes, for every real number. For a fair die,

F(4.7) = \frac{4}{6}

, because nothing is added between

4

and

5

, and

F(-3) = 0

. A CDF given by a formula needs the same care:

1 - e^{-x}

for a waiting time must be paired with

F(x) = 0

for

x < 0

, or it gives

F(-1) = 1 - e \approx -1.72

, a negative probability.Read more →

Can a CDF have no jumps and no density?

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Yes. Write

X

in base

3

with each digit a fair choice of

0

or

2

. Every single value has probability

0

, so the CDF never jumps, yet

X

always lies in the Cantor set, of total length

0

, so no density can describe it. Its CDF, the devil's staircase, climbs from

0

to

1

while staying flat on gaps of total length

1

inside

[0, 1]

.Read more →

Why is F(X) uniformly distributed?

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Because of how

F

accumulates. For a continuous

X

and any

u

between

0

and

1

, the event

F_X(X) \le u

means

X

has not passed the point where

F_X

reaches

u

, which happens with probability exactly

u

. So

F_X(X)

is uniform on

[0, 1]

. Run backwards,

F_X^{-1}(U)

for a uniform

U

has CDF

F_X

, a standard way to simulate any distribution.Read more →

Cumulative Distribution Function (CDF)

Accumulating Probability

Key Terms

From Events to Random Variables

CDF Defined (Core Meaning)

Mathematical Definition

CDF Notation

CDF Notation

Key Properties of the CDF

Discrete Random Variables

Continuous Random Variables

Mixed Distributions

Using the CDF to Compute Probabilities

CDF vs PMF vs PDF

Visual Interpretation

Common Mistakes

Why the CDF Matters

Connections to Other Probability Concepts

CDF Across the Three Cases

CDF FAQ

Why is the CDF defined with ≤ rather than <?

Does F(b) − F(a) include the endpoint a?

What is the difference between F(x) and f(x)?

What is the survival function?

Is the CDF defined for values the variable can never take?

Can a CDF have no jumps and no density?

Why is F(X) uniformly distributed?