Chapter 28: Expected Values of Continuous Random Variables

Meike Niederhausen and Nicky Wakim

2024-11-11

Learning Objectives

1. Calculate the 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 \(\mathbb{E}[X]\).

Integrating by Parts

\(\displaystyle\int_a^b u dv = uv\bigg|^b_a - \displaystyle\int_a^b vdu\)