Chapter 24: Continuous RVs and PDFs

Meike Niederhausen and Nicky Wakim

2023-11-08

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?

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

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

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

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. The solution is available in Meike’s video!

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. The solution is available in Meike’s video!

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