{"categories":["Probability Theory"],"contentHtml":"<p>For a discrete random variable, the probability mass function gives the mass at each possible value:</p>\n<p>$$p_X(x)=\\mathbb P(X=x),\\qquad \\sum_x p_X(x)=1.$$</p>\n<p>The CDF is a staircase formed by adding those masses. The notes use a die and the waiting time for a six to show how a geometric distribution arises: if \\(X\\) is the first roll on which a fair die shows six, then \\(\\mathbb P(X=n)=(5/6)^{n-1}(1/6)\\).</p>\n<p>Moments are sums in the discrete case. Absolute integrability gives</p>\n<p>$$\\mathbb E[X]=\\sum_x x\\,p_X(x),$$</p>\n<p>and variance measures the squared distance from the mean, \\(\\operatorname{Var}(X)=\\mathbb E[X^2]-\\mathbb E[X]^2\\). Linearity of expectation is the main calculation shortcut: it does not require independence.</p>","contentMarkdown":"For a discrete random variable, the probability mass function gives the mass at each possible value:\n\n$$p_X(x)=\\mathbb P(X=x),\\qquad \\sum_x p_X(x)=1.$$\n\nThe CDF is a staircase formed by adding those masses. The notes use a die and the waiting time for a six to show how a geometric distribution arises: if \\(X\\) is the first roll on which a fair die shows six, then \\(\\mathbb P(X=n)=(5/6)^{n-1}(1/6)\\).\n\nMoments are sums in the discrete case. Absolute integrability gives\n\n$$\\mathbb E[X]=\\sum_x x\\,p_X(x),$$\n\nand variance measures the squared distance from the mean, \\(\\operatorname{Var}(X)=\\mathbb E[X^2]-\\mathbb E[X]^2\\). Linearity of expectation is the main calculation shortcut: it does not require independence.","dataUrl":"https://sharifhsn.dev/api/posts/discrete-distributions-and-moments.json","date":"2024-09-23","datePublished":"2024-09-23","description":"For a discrete random variable, the probability mass function gives the mass at each possible value:","site":"https://sharifhsn.dev","slug":"discrete-distributions-and-moments","source":"FE-540 | Probability Theory","sourceUrl":null,"tags":["Probability Theory","Discrete Distributions","Moments"],"title":"Discrete Distributions and Moments","url":"https://sharifhsn.dev/blog/discrete-distributions-and-moments/","version":"1","wordCount":103}