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If $Y$ is continuous and $X$ is discrete with a finite number of points in the support, how to write the joint density of $(Y,X)$? For example, to write the joint density function evaluated at $(Y,X)=(y,x)$, is it true that I can only write it as $f(y|x)P_x$, where $f(\cdot|x)$ is the conditional density of $Y$ given $X=x$ and $P_x$ is the marginal pmf of $X=x$.

T34driver
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    In this instance, $X$ and $Y$ don’t _have_ a joint _density_ function in the usual meaning of the term and so there is no expression for the joint density. – Dilip Sarwate Sep 26 '20 at 14:38
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    Useful information can be found at https://stats.stackexchange.com/questions/91045, https://stats.stackexchange.com/questions/263506, https://stats.stackexchange.com/questions/134935, https://stats.stackexchange.com/questions/460261. One useful way to express such distributions is described, with a detailed example, at https://stats.stackexchange.com/questions/296385. – whuber Sep 26 '20 at 16:49
  • @DilipSarwate Thanks! Then is there a name for $f(y|x)P_x$? – T34driver Sep 27 '20 at 05:27
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    @whuber Thanks! The links are helpful. – T34driver Sep 27 '20 at 05:27

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