The FE-540 distribution notes connect several continuous families through the gamma function. The chi-squared family is a gamma distribution with parameters \\(n/2\\) and \\(1/2\\), which makes its moments and additivity properties easier to remember.
\nThe beta function normalizes densities on \\([0,1]\\):
\n$$\\mathrm B(a,b)=\\int_0^1x^{a-1}(1-x)^{b-1}\\,dx\n =\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}.$$
\nFor \\(X\\sim\\operatorname{Beta}(a,b)\\), the notes derive \\(\\mathbb E[X]=a/(a+b)\\) and \\(\\operatorname{Var}(X)=ab/((a+b)^2(a+b+1))\\). These identities are useful because they turn repeated integrals into parameter substitutions.
","contentMarkdown":"The FE-540 distribution notes connect several continuous families through the gamma function. The chi-squared family is a gamma distribution with parameters \\(n/2\\) and \\(1/2\\), which makes its moments and additivity properties easier to remember.\n\nThe beta function normalizes densities on \\([0,1]\\):\n\n$$\\mathrm B(a,b)=\\int_0^1x^{a-1}(1-x)^{b-1}\\,dx\n =\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a+b)}.$$\n\nFor \\(X\\sim\\operatorname{Beta}(a,b)\\), the notes derive \\(\\mathbb E[X]=a/(a+b)\\) and \\(\\operatorname{Var}(X)=ab/((a+b)^2(a+b+1))\\). These identities are useful because they turn repeated integrals into parameter substitutions.","dataUrl":"https://sharifhsn.dev/api/posts/gamma-beta-and-chi-squared-distributions.json","date":"2024-10-21","datePublished":"2024-10-21","description":"The FE-540 distribution notes connect several continuous families through the gamma function. The chi-squared family is a gamma distribution with parameters \\(n/2\\) and \\(1/2\\), wh…","site":"https://sharifhsn.dev","slug":"gamma-beta-and-chi-squared-distributions","source":"FE-540 | Probability Theory","sourceUrl":null,"tags":["Probability Theory","Gamma Distribution","Beta Distribution"],"title":"Gamma, Beta, and Chi-Squared Distributions","url":"https://sharifhsn.dev/blog/gamma-beta-and-chi-squared-distributions/","version":"1","wordCount":65}