Introduction to Statistics · Chapter 4 · Random Variables

Continuous Probability Functions

A probability stops being a table of rows and becomes an area under a curve — measured, not counted, and zero at any single point.

bookSHelf · Introduction to Statistics · §4.5 · a self-paced section

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Learning objectives — by the end of this section you will be able to

Objectives

Explain why probability for a continuous variable is area, not a list §4.5.1
State the two conditions a probability density function must satisfy definition
Calculate $P(a < x < b)$ for a flat density as the area of a rectangle §4.5.1
Explain why $P(x=c)$ is zero for any single value $c$ §4.5.1
Use the CDF $P(X \leq x)$ and the complement rule $P(X>x)=1-P(X<x)$ §4.5.2

4.5

§4.5.1 — Probability Is Area

You cannot write a row for every possible value of a continuous random variable — there are infinitely many. A curve solves that: it assigns probability to stretches of the number line, something you can measure, instead of to points.

A probability is the area under a curve — never a number you read straight off it.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

§4.5.1 — the two-part test any density has to pass

Probability Density Function (PDF)

Definition 4.5.1 — Probability Density Function

A probability density function for a continuous random variable $X$ is a function $f(x)$ with two properties:

$f(x) \geq 0$ for every value of $x$ .
The total area between the graph of $f(x)$ and the x-axis is exactly one.

Figure: the whole area under the density is 1, and a slice of it is a probability.

The probability that $X$ lands between $a$ and $b$ is the area under $f(x)$ from $a$ to $b$ : $P(a < x < b) = \text{area under } f(x) \text{ from } a \text{ to } b$

Continuous Probability Functions · bookSHelf Intro Stats§4.5

§4.5.1 — a height is not a probability

$f(x)$ is a density, not a probability

The height of the curve at $x=3$ is not "the probability of 3" — it is a density, a rate of probability per unit of $x$ . It only becomes a probability once you multiply it by a width.

For a flat density, that multiplication is just base × height — a rectangle. If an area comes out bigger than 1, or negative, you've made an arithmetic mistake: no region can exceed the whole unit of area, and no width runs backwards.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Context Pause — why a curve replaces a table

A curve assigns probability to stretches, not points.

There are infinitely many possible values, and any list you start leaves out the number halfway between the first two entries. A stretch of the number line is something you can measure; a single point is not.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Insight Note — every area in this section is a rectangle

Base times height. That is the whole computation.

When $f(x)$ is a flat horizontal line, the region under it is a rectangle — no calculus, no formula sheet. Multiply the width of the interval you care about by the height of the line, and you have the probability.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Your turn — check both properties, in order

Try It Now 4.5.1

Try It Now 4.5.1

Consider $f(x) = \frac{1}{8}$ for $0 \leq x \leq 8$ , and $f(x) = 0$ everywhere else. Show that $f(x)$ is a legitimate probability density function.

Both properties hold

1. On $0 \leq x \leq 8$ , $f(x) = \frac{1}{8} > 0$ ; everywhere else it is 0 — never negative.

2. $\text{AREA} = 8\left(\frac{1}{8}\right) = 1$

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Worked example — confirm the density integrates to one

Example 4.5.1a: The Total Area Is One

Example 4.5.1 — Areas Under a Flat Density

Consider $f(x) = \frac{1}{20}$ for $0 \leq x \leq 20$ . a. Show that the total area under $f(x)$ is one.

The graph is a horizontal segment at height $\frac{1}{20}$ , so the region under it is a rectangle with base 20:

$\text{AREA} = 20\left(\frac{1}{20}\right) = 1$

Figure 4.5.1 - The density f(x) = 1/20 drawn as a flat segment over its whole support.

Figure 4.5.1: the density $f(x)=\frac{1}{20}$ drawn as a flat segment over its whole support.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Worked example — a narrow strip at the left edge

Example 4.5.1b: $P(0 < x < 2)$

Example 4.5.1, continued

b. Find $P(0 < x < 2)$ . The region is again a rectangle — base from 0 to 2, height still $\frac{1}{20}$ .

$\text{AREA} = (2-0)\left(\frac{1}{20}\right) = 0.1$

So $P(0 < x < 2) = 0.1$ .

Figure 4.5.2 - The strip from x = 0 to x = 2 has area (2)(1/20) = 0.1.

Figure 4.5.2: the strip from $x=0$ to $x=2$ has area $(2)\left(\frac{1}{20}\right) = 0.1$ .

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Worked example — same move, wider base

Example 4.5.1c: $P(4 < x < 15)$

Example 4.5.1, continued

c. Find $P(4 < x < 15)$ . The base is $(15-4)=11$ .

$\text{AREA} = (15-4)\left(\frac{1}{20}\right) = 0.55$

So $P(4 < x < 15) = 0.55$ .

Figure 4.5.3 - The strip from x = 4 to x = 15 has area (11)(1/20) = 0.55.

Figure 4.5.3: the strip from $x=4$ to $x=15$ has area $(11)\left(\frac{1}{20}\right) = 0.55$ .

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Worked example — a single point has no width

Example 4.5.1d: $P(x = 15)$

Example 4.5.1, continued

d. Find $P(x = 15)$ . On the graph, $x=15$ is a vertical line — a vertical line has no width, so its base is zero.

$P(x=15) = (\text{base})(\text{height}) = (0)\left(\frac{1}{20}\right) = 0$

Not a quirk of this function — for any continuous random variable, the probability of landing on one exact value is zero.

Figure 4.5.4: a single value is a line with no width, so $P(x=15)=0$ .

4.5

§4.5.2 — The Cumulative Distribution Function

Rather than re-measuring a rectangle every time, tabulate the area to the left of each point once. Every "less than," "greater than," or "between" question becomes a subtraction on numbers you already have.

The area to the left of $x$ — tabulated once.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

§4.5.2 — the running total behind every question

Cumulative Distribution Function (CDF)

Definition 4.5.2 — Cumulative Distribution Function (CDF)

The cumulative distribution function of a continuous random variable $X$ is

$P(X \leq x) = \text{the area under } f(x) \text{ to the left of } x$

The complement: $P(X > x) = 1 - P(X < x)$

Figure 4.5.5: the cdf is the area to the left; whatever is left over is the area to the right.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

§4.5.2 — reading the split

The shaded block is $P(X<x)$ ; the rest is $P(X>x)$

Together the two blocks are the entire rectangle, whose area is 1 — so the two probabilities always add to 1, exactly what the complement rule says.

Because a single point contributes no area, $P(X \leq x)$ and $P(X < x)$ are the same number for a continuous distribution — the endpoint contributes nothing either way.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Context Pause — one running total answers every question

Tabulate the area to the left once; every question becomes subtraction.

Rather than re-measuring a rectangle each time, compute the area to the left of each point once. Every "less than," "greater than," or "between" question is then a subtraction on numbers you already have.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Your turn — draw it first, then measure the strip

Try It Now 4.5.2

Try It Now 4.5.2

Consider $f(x) = \frac{1}{8}$ for $0 \leq x \leq 8$ . Draw the graph of $f(x)$ , scale both axes, shade the region between $x=2.5$ and $x=7.5$ , and find $P(2.5 < x < 7.5)$ .

Base 5, height $\frac{1}{8}$

$P(2.5 < x < 7.5) = (7.5-2.5)\left(\frac{1}{8}\right) = 0.625$

Check: 0.625 is between 0 and 1, and the shaded strip covers a bit more than half the rectangle — consistent.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Worked example — a strip between two interior values

Example 4.5.2: A Probability Between Two Interior Values

Example 4.5.2

Return to $f(x) = \frac{1}{20}$ on $0 \leq x \leq 20$ . Shade the region between $x=2.3$ and $x=12.7$ and find $P(2.3 < x < 12.7)$ .

Height $\frac{1}{20}$ , base $12.7-2.3=10.4$ :

$P(2.3 < x < 12.7) = (12.7-2.3)\left(\frac{1}{20}\right) = 0.52$

Figure 4.5.6: the strip between $x=2.3$ and $x=12.7$ shaded under the flat density.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Try it in rāSHio — read the shaded area straight off the tool

Distributions → Normal shades the area for you

Open rāSHio and choose Distributions → Normal. Type the value you're asking about and the tool shades the area to its left, reading the probability off directly.

That shaded area is $P(X \leq x)$ , and whatever stays unshaded is $P(X > x)$ . The density in this section is flat, so you can check it with a ruler — the tool runs the same left-area / right-area split on a curve you cannot.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

§4.5.2 — watching the split move

rāSHio: Distributions → Normal

Figure 4.5.7: reading area to the left as a probability in rāSHio — Distributions → Normal.

Continuous Probability Functions · bookSHelf Intro Stats§4.5

Key Terminology — the four terms this section defined

The vocabulary

The variable

continuous random variable — a quantity whose possible values fill an unbroken stretch of the number line, so that between any two values there is always another.
uniform density — a density that is a flat horizontal line over its interval, so every stretch of equal width carries equal probability.

Its distribution

probability density function (pdf) — a non-negative function $f(x)$ whose total area with the x-axis is one, and whose area between two values gives the probability the variable lands between them.
cumulative distribution function (cdf) — $P(X \leq x)$ , the area under $f(x)$ to the left of $x$ .

4.5

The headline result of §4.5

Probability is area — and a single point has none

Every question in this section reduces to one multiplication: base × height. A single value, being a line with no width, always contributes exactly zero.

$P(x=c) = (0)(\text{height}) = 0 \text{ for any continuous } X \text{ and any } c$

† That is why $P(X < x)$ and $P(X \leq x)$ are the same number for a continuous variable — the endpoint contributes nothing.

4.5

Continuous Probability Functions · bookSHelf Intro Stats§4.5

§4.5 — conclusions

What §4.5 leaves you with

The core idea

Probability is area under a density. For a flat density that area is always base × height, the total area is always 1, and the cdf $P(X \leq x)$ tabulates the area to the left so every other question becomes a subtraction.

The failure case

Treating $f(x)$ itself as a probability, or forgetting that a single exact value is a line with zero width — and therefore zero probability, no matter how likely the value "feels."

Next: §4.6 — Continuous Distribution, the general case where the density is no longer flat. Back to start.

Continuous Probability Functions

Objectives

Probability Density Function (PDF)

f(x)f(x)f(x) is a density, not a probability

Try It Now 4.5.1

Example 4.5.1a: The Total Area Is One

Example 4.5.1b: P(0<x<2)P(0 < x < 2)P(0<x<2)

Example 4.5.1c: P(4<x<15)P(4 < x < 15)P(4<x<15)

Example 4.5.1d: P(x=15)P(x = 15)P(x=15)

Cumulative Distribution Function (CDF)

The shaded block is P(X<x)P(X<x)P(X<x); the rest is P(X>x)P(X>x)P(X>x)

Try It Now 4.5.2

Example 4.5.2: A Probability Between Two Interior Values

Distributions → Normal shades the area for you

rāSHio: Distributions → Normal

The vocabulary

What §4.5 leaves you with

$f(x)$ is a density, not a probability

Example 4.5.1b: $P(0 < x < 2)$

Example 4.5.1c: $P(4 < x < 15)$

Example 4.5.1d: $P(x = 15)$

The shaded block is $P(X<x)$ ; the rest is $P(X>x)$