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

What is a cumulative distribution function?

Cumulative distribution function

The cumulative distribution function (cdf) of a continuous random variable \(X\), is the function \(F_X(x)\), such that for all real values of \(x\), \[F_X(x)= \mathbb{P}(X \leq x) = \int_{-\infty}^x f_X(s)ds\]

Remarks: In general, \(F_X(x)\) is increasing and

  • \(\lim_{x\rightarrow -\infty} F_X(x)= 0\)

  • \(\lim_{x\rightarrow \infty} F_X(x)= 1\)

  • \(P(X > a) = 1 - P(X \leq a) = 1 - F_X(a)\)

  • \(P(a \leq X \leq b) = F_X(b) - F_X(a)\)

Let’s demonstrate the CDF with an example

Example 2

Let \(f_X(x)= 2\), for \(2.5 \leq x \leq 3\). Find \(F_X(x)\).

Derivatives of the CDF

Theorem 1

If \(X\) is a continuous random variable with pdf \(f_X(x)\) and cdf \(F_X(x)\), then for all real values of \(x\) at which \(F'_X(x)\) exists, \[\frac{d}{dx} F_X(x)= F'_X(x) = f_X(x)\]

Finding the PDF from a CDF

Example 3

Let \(X\) be a RV with cdf \[F_X(x)= \left\{ \begin{array}{ll} 0 & \quad x < 2.5 \\ 2x-5 & \quad 2.5 \leq x \leq 3 \\ 1 & \quad x > 3 \end{array} \right.\] Find the pdf \(f_X(x)\).

Let’s go through another example (1/7)

Example 4

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

  1. Show \(f_X(x)\) is a pdf.

  2. Find \(\mathbb{P}(1 \leq X \leq 3)\).

  3. Find \(F_X(x)\).

  4. Given \(F_X(x)\), find \(f_X(x)\).

  5. Find \(\mathbb{P}(X \geq 1 | X \leq 3)\).

  6. Find the median of the distribution of \(X\).

Let’s go through another example (2/7)

Example 4.1

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

  1. Show \(f_X(x)\) is a pdf.

Let’s go through another example (3/7)

Do this problem at home for extra practice.

Example 4.2

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

  1. Find \(\mathbb{P}(1 \leq X \leq 3)\).

Let’s go through another example (4/7)

Example 4.3

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

  1. Find \(F_X(x)\).

Let’s go through another example (5/7)

Do this problem at home for extra practice.

Example 4.4

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

  1. Given \(F_X(x)\), find \(f_X(x)\).

Let’s go through another example (6/7)

Example 4.5

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

  1. Find \(\mathbb{P}(X \geq 1 | X \leq 3)\).

Let’s go through another example (7/7)

Example 4.6

Let \(X\) be a RV with pdf \(f_X(x)= 2e^{-2x}\), for \(x>0\).

  1. Find the median of the distribution of \(X\).

Chapter 24 Slides