From fca2ee8693515fbcec69ad0adc128dfc3e4d1282 Mon Sep 17 00:00:00 2001
From: JellyBoonz <86168103+JellyBoonz@users.noreply.github.com>
Date: Mon, 3 Mar 2025 23:00:26 -0500
Subject: [PATCH] Change inequalities for Variance Bias

The second and third "<" should actually be "=" because of the unbiasedness of the sample variance
---
 03-jensens-inequality.Rmd | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/03-jensens-inequality.Rmd b/03-jensens-inequality.Rmd
index 0c50c91..a834de1 100644
--- a/03-jensens-inequality.Rmd
+++ b/03-jensens-inequality.Rmd
@@ -92,8 +92,8 @@ The bias in the estimate of the sample mean's standard error originates from the
 $$
 \begin{aligned}
 \mathbb{E}\left[\hat{\sigma}\right] = \mathbb{E}\left[\sqrt{\hat{V}(\bar{X})}\right] &< \sqrt{\mathbb{E}[\hat{V}(\bar{X})]} \;\;\; \text{(by Jensen's Inequality)}\\
-& < \sqrt{V(\bar{X})} \;\;\; \text{(since the sampling variance is unbiased)} \\
-& < \sigma. \;\;\; \square
+& = \sqrt{V(\bar{X})} \;\;\; \text{(since the sampling variance is unbiased)} \\
+& = \sigma. \;\;\; \square
 \end{aligned}
 $$