https://www.chenk.top/en/tags/expectation/Recent content in Expectation on Chen Kai BlogHugoenWed, 21 Aug 2024 09:00:00 +0000https://www.chenk.top/en/probability-statistics/03-expectation-and-moments/Wed, 21 Aug 2024 09:00:00 +0000https://www.chenk.top/en/probability-statistics/03-expectation-and-moments/<p>A probability distribution is a complete description of a random variable — it tells you the probability of every possible outcome. But complete descriptions are unwieldy. When someone asks &ldquo;how tall are people in this city?&rdquo;, you don&rsquo;t hand them a density function; you say &ldquo;about 170 cm on average, give or take 10 cm.&rdquo; The average and the spread capture most of what matters in practice.</p> <p>This article develops the mathematical framework for summarizing distributions. We start with expectation (the center), build up to variance (the spread), and then introduce moment-generating functions — a single formula that encodes every moment of a distribution and, remarkably, uniquely determines the distribution itself.</p>