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Say I have one PDF given by $$ f(r) = \begin{cases} \frac{3 r^{2}}{8}, \text{if } 0 \leq r \leq 2\\ 0\text{, otherwise} \end{cases} $$ describing the distribution of a continuous random variable $R$ due to, for example, some external reasons, and another one given by $$ g(r) = \begin{cases} \frac{r}{2}, \text{if } 0 \leq r \leq 2\\ 0\text{, otherwise} \end{cases} $$ describing, for example, some degeneracy in the system (larger values of $R$ are more likely).

I with to obtain a new PDF defined on $r \in [0, 2]$ which would weight the probability density values given by $f(r)$ using the corresponding relative probabilities of $r$ given by $g(r)$. I know that $$ f(r) \times g(r) = \frac{3r^{3}}{16} $$ is not what I want because $$ \int_{0}^{2} \frac{3r^{3}}{16} \text{d}r = \frac{3}{4} $$ but it should be $1$ if it were a PDF.

On the other hand, $$ h(r) = \frac{f(r) \times g(r)}{\int_{0}^{2} f(r) \times g(r) \text{d}r} = \frac{r^{3}}{4} $$ seems like a valid PDF because $$ \int_{0}^{2}\frac{r^{3}}{4}\text{d}r = 1 $$

Problem is, I've found that equations using normalization like that in $h(r)$ always seem like valid PDFs, as they integrate to $1$ over the interval used in the denominator. For example, $$ h^{*}(r) = \frac{f(r) \times g(r) \times g(r)}{\int_{0}^{2} f(r) \times g(r) \times g(r) \text{d}r} = \frac{5 r^{4}}{32} $$ for $r \in [0,2]$ could be a valid PDF too.

My question is: is the PDF in the form of $h(r)$ the one I'm looking for ? In my research in adding PDFs I've encountered terms like convolution or product distribution, with which I'm not familiar yet, and the corresponding formulas seem more involved than what I'm showing here, but I can't judge with my current knowledge whether they are applicable to my problem.

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  • $\begingroup$ So it sounds like you have $P(R|External)$ and $P(R)$. The relation would then be $P(R)=\int P(R|External)dP(External)$. It's not clear what you mean by combining the two because you already have $P(R)$. $\endgroup$ – Alex R. Mar 2 '17 at 18:50
  • $\begingroup$ What I need is to scale one of the PDFs (say $f(r)$) using the relative probabilities of individual $r$s given by the other PDF ($g(r)$) so that the final function is a PDF too (integrates to $1$ over the interval where $f(r) \text{and} g(r)$ are positive). I've edited the "external" parts so that it's more clear that it was meant as an example. $\endgroup$ – matlab-oh-no Mar 2 '17 at 19:04

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