Lesson 8: Probability density functions (PDFs)

Meike Niederhausen and Nicky Wakim

2025-10-20

Learning Objectives

1. Distinguish between discrete and continuous random variables.

2. Calculate probabilities for continuous random variables.

3. Use R to simulate known continuous distributions.

Where are we?

Learning Objectives

1. Distinguish between discrete and continuous random variables.

2. Calculate probabilities for continuous random variables.

3. Use R to simulate known continuous distributions.

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 \(x\)

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

Learning Objectives

1. Distinguish between discrete and continuous random variables.

2. Calculate probabilities for continuous random variables.

3. Use R to simulate known continuous distributions.

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 \(a \leq x \leq 3\).

2. 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 \(a \leq x \leq 3\).

3. 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 \(a \leq x \leq 3\).

4. 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 \(a \leq x \leq 3\).

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

Learning Objectives

1. Distinguish between discrete and continuous random variables.

2. Calculate probabilities for continuous random variables.

3. Use R to simulate known continuous distributions.

Use R to simulate known distributions

• We can use R to simulate continuous random variables and visualize their distributions
• For example, we can simulate a uniform distribution between 2.5 and 3

uniform = tibble(
  x = runif(n=10000, min=2.5, max=3)
)

ggplot(uniform, 
       aes(x = x, 
           y = after_stat(density))) +
  geom_histogram( binwidth = 0.001) + 
  geom_abline(intercept = 2, slope = 0) + 
  labs(
    title = "Probability density function (pdf) of X",
    x = "x",
    y = "pdf"
  ) 

Use R to simulate any continuous distribution

• We will discuss other ways to simulate continuous distributions once we cover cumulative distribution functions (CDFs) and inverse CDFs