Lecture 9

Continuous Distributions


Grace Tompkins

Last modified — 23 Oct 2026

Learning Outcomes

By the end of this lecture, students are anticipated to be able to:

  • Define a probability density function
  • Use the continuous PDFs to solve for probabilities

1 Continuous Random Variables

Continuous Random Variables

A RV \(X\) is continuous if \[\mathbb{P}(X=x) = 0,\] for every \(x\in{\mathbb{R}}\).

  • This is the most rigorous way to define continuous RVs, but it doesn’t provide much intuition.

  • We need a bit more to make our intuition match the math.

Absolutely Continuous RVs and Density Functions

A function \(f : {\mathbb{R}}\to {\mathbb{R}}\) is called a density function if \(f(x)\geq 0,\ \forall x\in{\mathbb{R}}\) and \(\int f(x)\mathsf{d}x=1\).

A RV \(X\) is absolutely continuous if there exists a density function \(f\) such that \[\mathbb{P}(a\leq X\leq b) = \int_a^b f(x)\mathsf{d}x\] whenever \(a\leq b\).

We call such a density function a probability density function (PDF) and will often use the notation \(f_X(x)\) to denote its relationship to \(X\).

Probability of \(a \le X = b\) visually

Absolutely Continuous RVs

Let \(X\) be an absolutely continuous random variable. Then \(X\) is a continuous random variable (i.e., \(\mathbb{P}(X = a) = 0\) for all \(a \in {\mathbb{R}}\))

\(\mathbb{P}(X = a) = \mathbb{P}(a \le X \le a) = \int_a^a f(x) dx = 0\)

Note

Note: the converse is not necessarily true. That is, not all continuous distributions you will come across in statistics (in general) are absolutely continuous. We will stick to absolutely continuous distributions in this course.

Probability of \(X = a\) visually

Comparison to Discrete RVs

  • Consider some set \(A \subset {\mathbb{R}}\).

  • Let \(X\) be a random variable.

Probability of \(A\) for discrete RV:

\[\mathbb{P}(X \in A) = \sum_{x \in A} p_X(x).\]

Probability of \(A\) for an (absolutely) continuous RV:

\[\mathbb{P}(X \in A) = \int_{A} f_X(x) \mathsf{d}x.\]

Continuous PDFs

Let \(X\) be a RV with PDF \[f_X(x) = 2 x^{-3} I_{[1,\infty)}(x).\] a. Is \(f\) a valid density function?

  1. What is \(\mathbb{P}(X = 2)\)?

  2. What is \(\mathbb{P}(0<X<2)\)?

  3. Suppose we defined \(g(x) = 2 x^{-3}\) (no indicator function). Is \(g(x)\) a density function? Does it have the same support?

Continuous PDFs

Continuous PDFs

Let \(f_X(x) = \frac{1}{5}I_{[0,5]}(x)\).

  1. What is \(\mathbb{P}(X \le 3)\)?

  2. What is \(\mathbb{P}(X < 3)\)?

  3. What is \(\mathbb{P}(-4 < X < 10)\)?

  4. What is \(\mathbb{P}(X \ge 2)\)?

Continuous PDFs

Reminders

  • No class on Monday (thanksgiving)

  • Midterm 1 on Friday October 16th (all material up to today’s class)

  • Start prepping your cheat sheet

    • MUST BE HAND WRITTEN! No photo copies, typed, or tablet-written and printed sheets allowed
    • 8.5 inches x 11 inches or less
    • You can habe one single page, and can write on both sides

Checkpoint 3

You may have:

  • HANDWRITTEN Cheat Sheet

  • Non-graphing calculator

  • Writing materials

Phones and laptops put away, please! You can have 15 minutes.