Chapter 24: Continuous RVs and PDFs

Meike Niederhausen and Nicky Wakim

2024-11-06

Learning Objectives

  1. Distinguish between discrete and continuous random variables.

  2. Calculate probabilities for continuous random variables.

  3. Calculate and graph a density (i.e., probability density function, PDF).

  4. Calculate and graph a CDF (i.e., a cumulative distribution function)

Discrete vs. Continuous RVs

  • For a discrete RV, the set of possible values is either finite or can be put into a countably infinite list.

     

  • Continuous RVs take on values from continuous intervals, or unions of continuous intervals

Figure from Introduction to Probability TB (pg. 301)

How to define probabilities for continuous RVs?

Discrete RV \(X\):

  • pmf: \(p_X(x) = P(X=x)\)

Continuous RV \(X\):

  • density: \(f_X(x)\)
  • probability: \(P(a \leq X \leq b) = \int_a^b f_X(x)dx\)

What is a probability density function?

Probability density function

The probability distribution, or probability density function (pdf), of a continuous random variable \(X\) is a function \(f_X(x)\), such that for all real values \(a,b\) with \(a \leq b\),

\[\mathbb{P}(a \leq X \leq b) = \int_a^b f_X(x)dx\]

Remarks:

  1. Note that \(f_X(x) \neq \mathbb{P}(X=x)\)!!!

  2. In order for \(f_X(x)\) to be a pdf, it needs to satisfy the properties

    • \(f_X(x) \geq 0\) for all \(x\)

    • \(\int_{-\infty}^{\infty} f_X(x)dx=1\)

Let’s demonstrate the PDF with an example (1/5)

Example 1.1

Let \(f_X(x)= 2\), for \(a \leq x \leq 3\).

  1. Find the value of \(a\) so that \(f_X(x)\) is a pdf.

Let’s demonstrate the PDF with an example (2/5)

Example 1.2

Let \(f_X(x)= 2\), for \(a \leq x \leq 3\).

  1. Find \(\mathbb{P}(2.7 \leq X \leq 2.9)\).

Let’s demonstrate the PDF with an example (3/5)

Example 1.3

Let \(f_X(x)= 2\), for \(a \leq x \leq 3\).

  1. Find \(\mathbb{P}(2.7 < X \leq 2.9)\).

Let’s demonstrate the PDF with an example (4/5)

Example 1.4

Let \(f_X(x)= 2\), for \(a \leq x \leq 3\).

  1. Find \(\mathbb{P}(X = 2.9)\).

Let’s demonstrate the PDF with an example (5/5)

Example 1.5

Let \(f_X(x)= 2\), for \(a \leq x \leq 3\).

  1. Find \(\mathbb{P}(X \leq 2.8)\).