From single-consumer-multi-units to multiple-consumers-single-unit
The cumulative distribution function (CDF) describes the probability that a random variable X with a given probability distribution will be found to have a value less than or equal to x. That is: $$F_X(x) = P(X \leq x) $$

In this context, think of $X$ as the valuation of a particular unit demander. The demands are distributed with a certain probability distribution. So the the CDF tells you the probability of  a randomly drawn demander of having a valuation at least as large as $x$. If everyone demands either one unit if the price $p \leq X$ and 0 otherwise, then this also gives a demand function:

$$Q_d(p) = P(X \geq p) = 1 - P(X \leq p) $$

because at each price $p$ everyone with a valuation at least as high as $p$ will buy their one unit. But the fraction of people with a valuation at least as high as $p$ is one minus the fraction of people with valuations $p$ or lower. 

Next, consider a single household with a demand function that just so happens to look like $1 - P(X \leq p) $. This household will have an identical demand for the good at all prices as the collective demand of the unit demanders above. 

I believe this generalizes to the more complex case you present in the question but I haven't worked out the details.