I'm trying so hard to clear out my simple confusion but I'm getting nowhere near to it :(
Let's say, the PDF for weight measurement of all adults (age $\ge$ 18) is given $\large p_{adults}$, using this, I can tell what's the probability that an adult person would weigh between a Kgs to b Kgs...
And the PDF for weight measurement of all non-adults (age $\lt $ 18) is given $\large p_{non-adults}$, and using this, I can tell what's the probability that a non-adult person would weigh between c Kgs to d Kgs...
Since all humans is either an adult or not... can't I find a PDF that would represent the weight measurement of all humans? Or in other words, what would be a single PDF such that can cover all the cases of $\large p_{adults}(x)$ and $\large p_{non-adults}(x)$?
If this is possible to find then how?? If not then why not??
I am thinking that the answer should be yes, and that single PDF might be represented by some operation between $\large p_{adults}(x)$ and $\large p_{non-adults}(x)$, like addition or multiplication, or maybe convolution or something like that... what would that be?
Use the law of total probability. Let $F_A$ and $F_C$ be the cumulative distribution functions for the weights of adults and children, respectively. If the proportion of adults in the population is $p$, then \begin{align*} \mathbb P(X\leq x)&=\mathbb P(X\leq x\mid\text{Adult})\cdot\mathbb P(\text{Adult})+\mathbb P(X\leq x\mid\text{Child})\cdot\mathbb P(\text{Child})\\ &=pF_A(x)+(1-p)F_C(x). \end{align*} Differentiating implies that the density is given by $f(x)=pf_A(x)+(1-p)f_C(x)$.
This is a relatively simple example of mixture distributions.