Hi,
When you activate the random option, the randomness it calculated each time you display a page of the pagination.
If we allowed you to have random and infinite scroll working together, you would get random elements and scrolling down, you would get again more elements which would be also selected randomly among all the products of the listing, so you'll see some products several times and you won't see some products.
That's not good for your customers nor for your SEO which is why we disable the infinite scroll when the random option is activated.
If you really want to do it, you can change the line:
$infinite_scroll = !$enableCarousel && ((int)$this->params->get('infinite_scroll', 0) == 1) && ((int)$this->params->get('random', 0) == 0);
to:
$infinite_scroll = !$enableCarousel && ((int)$this->params->get('infinite_scroll', 0) == 1);
in the file product / listing_div.php via the menu Display>Views.