Notes on the Central Limit Theorem

<h1>Theory</h1> <p>Let&nbsp;<em>Xᵢ</em>&nbsp;with&nbsp;<em>i</em>&nbsp;= {1 &hellip;&nbsp;<em>n</em>} be&nbsp;<em>n</em>&nbsp;random variables (r.v.), independently distributed with distributions D<em>ᵢ</em>&nbsp;not necessarily identical, but with the same mean&nbsp;<em>&mu;</em>&nbsp;and variance&nbsp;<em>&sigma;</em>&sup2;.</p> <p><img alt="" src="https://miro.medium.com/v2/resize:fit:383/1*I43WcRXtsNw5gdVuC26NmQ.png" style="height:40px; width:383px" /></p> <p>(1)</p> <p>Let us define the &ldquo;<em>sum</em>&rdquo; r.v.&nbsp;<em>Sₙ</em></p> <p><img alt="" src="https://miro.medium.com/v2/resize:fit:456/1*6rcLsvDWAqBL-1_QBJoqdw.png" style="height:86px; width:456px" /></p> <p>(2)</p> <p>By the linearity property of the expected value</p> <p><a href="https://medium.com/@m.pierini/notes-on-central-limit-theorem-13e4ba94f08c"><strong>Learn More</strong></a></p>
Tags: Limit theorem