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my question is quite generic. I am currently studying the algorithms calculating random numbers from distributions:

In inverse transform method we get the cumulative distribution function in the end and take the random variable from there. Whereas in rejection method, we are working directly with pdf. I got a little bit confused if we normally sample from cdf , since I have never come across it before.

I know what the cdf and pdf are and that they are tightly linked.

Hopefully, someone can clarify it to me if sampling from cdf is a common approach.

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    $\begingroup$ There are a very large number of ways to sample from a distribution. Many do not need mathematical representations of the PDF or the CDF. It's difficult to use the CDF directly, but its inverse -- the quantile function -- gives perhaps the simplest, most direct method of sampling provided the quantile function can be computed. For more about this particular method see stats.stackexchange.com/search?q=inverse+transform+sampling. $\endgroup$
    – whuber ♦
    Commented Nov 12, 2020 at 17:12
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    $\begingroup$ Theoretically one can simulate from any function identifying the distribution, e.g., pdf, cdf, mgf, quantile function, characteristic function... $\endgroup$
    – Xi'an
    Commented Nov 12, 2020 at 17:26

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There are algorithms that sample random variates using both the PDF and CDF of a distribution. One example is the inversion-rejection method described in chapter 7 of Non-Uniform Random Variate Generation (Devroye 1986), which works for any unimodal distribution for which the mode is known.

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