{"id":420,"date":"2015-01-05T20:05:34","date_gmt":"2015-01-06T02:05:34","guid":{"rendered":"http:\/\/frees.pajarel.net\/?page_id=420"},"modified":"2015-02-20T19:09:00","modified_gmt":"2015-02-21T01:09:00","slug":"4-2-inverse-transform-image-inversedf-needed","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/actuarial-mathematics\/interest-rate-risks-and-simulation\/4-simulation\/4-2-inverse-transform-image-inversedf-needed\/","title":{"rendered":"4.2 Inverse Transform"},"content":{"rendered":"

With the sequence of uniform random numbers, we next transform them to a distribution of interest. Let \\(F\\) represent a distribution function of interest. Then, use the inverse transform<\/em>
\n\\[
\nX_i=F^{-1}\\left( U_i \\right) .
\n\\]
\nThe result is that the sequence {\\(X_{i}\\)} is approximately i.i.d. with distribution function \\(F\\).<\/p>\n

To interpret the result, recall that a distribution function, \\(F\\), is monotonically increasing and so the inverse function, \\(F^{-1}\\), is well-defined. The inverse distribution function (also known as the quantile function<\/em>), is defined as
\n\\begin{eqnarray*}
\nF^{-1}(y) = \\inf_x { F(x) \\ge y } ,
\n\\end{eqnarray*}
\nwhere “\\(inf\\)” stands for “infinum”, or the greatest lower bound.<\/p>\n

Here is a graph to help you visualize the inverse transform. When the random variable is continuous, the distribution function is strictly increasing and we can readily identify a unique inverse at each point of the distribution.<\/p>\n

\"InverseDF\"<\/a>

<\/figcaption><\/figure>\n

The inverse transform result is available when the underlying random variable is continuous, discrete or a mixture. Here is a series of examples to illustrate its scope of applications.<\/p>\n

Exponential Distribution Example.<\/strong> Suppose that we would like to generate observations from an exponential distribution with scale parameter \\(\\theta\\) so that \\(F(x) = 1 – e^{-x\/\\theta}\\). To compute the inverse transform, we can use the following steps:
\n\\begin{eqnarray*}
\ny = F(x) &\\Leftrightarrow& y = 1-e^{-x\/\\theta} \\
\n&\\Leftrightarrow& -\\theta \\ln(1-y) = x = F^{-1}(y) .
\n\\end{eqnarray*}
\nThus, if \\(U\\) has a uniform (0,1) distribution, then \\(X = -\\theta \\ln(1-U)\\) has an exponential distribution with parameter \\(\\theta\\).<\/p>\n

Some Numbers.<\/em> Take \\(\\theta = 10\\) and generate three random numbers to get<\/p>\n\n\n\n
\\(U\\)<\/td>\n0.26321364<\/td>\n0.196884752<\/td>\n0.897884218<\/td>\n<\/tr>\n
\\(X = -10\\ln(1-U)\\)<\/td>\n1.32658423<\/td>\n0.952221285<\/td>\n<\/td>\n<\/tr>\n<\/table>\n

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With the sequence of uniform random numbers, we next transform them to a distribution of interest. Let \\(F\\) represent a distribution function of interest. Then, use the inverse transform \\[ X_i=F^{-1}\\left( U_i \\right) . \\] …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":415,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-6M","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/420"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=420"}],"version-history":[{"count":7,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/420\/revisions"}],"predecessor-version":[{"id":1645,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/420\/revisions\/1645"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/415"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}