Uploaded July 2025 | Updated September 2026, 2 weeks ago

---

Here we use the moment generating function to prove the central limit theorem. This is one of the most important results in probability, and the proof provides deep insight into why it is true.

This video was produced at the University of Washington, and we acknowledge funding support from the Boeing Company

%%% CHAPTERS %%%
00:00 Intro
00:51 Formal Statement of the CLT
03:48 Equivalent Statements of the CLT
05:50 Step 1: Equivalence via MGFs
08:20 Step 2: MGF of Sample Mean wrt MGF of X
10:24 Step 3: Taylor Expansion
13:28 Step 4: Normalization Terms in MGF
16:10 Step 5: Normalization Terms in Taylor Expansion
18:11 Step 6: Taking the Limit
20:38 Reviewing the Proof
25:35 Outro