Chapter 28: Expected Values of Continuous Random Variables

Meike Niederhausen and Nicky Wakim

2023-11-20

Table of contents

• Learning Objectives
• Expected value of a function of a continuous RV
• Expected Value of the Uniform Distribution
• Expected Value of the Exponential Distribution
• Expected value from a joint distribution

Learning Objectives

1. Calculated mean (expected value) of a continuous RV

Expected value of a function of a continuous RV

How do we calculate expected values of discrete RVs?

       

For discrete RVs: weight average \[\mathbb{E}[X] = \sum_{i=1}^n x_ip_X(x_i).\]

How do we calculate expected values of continuous RVs?

       

For continuous RVs:

Expected Value of the Uniform Distribution

Example 1

Let \(f_X(x)= \frac{1}{b-a}\), for \(a \leq x \leq b\). Find \(\mathbb{E}[X]\).

Expected Value of the Exponential Distribution

Example 2

Let \(f_X(x)= \lambda e^{-\lambda x}\), for \(x > 0\) and \(\lambda> 0\). Find \(\mathbb{E}[X]\).

Expected value from a joint distribution

Example 3

Let \(f_{X,Y}(x,y)= 2e^{-(x+y)}\), for \(0 \leq x \leq y\). Find \(\mathbb{E}[X]\).