Why don't my AdSense impressions equal my page views?

You ask a good question because while Google does explain the stats shown on a Google AdSense report, it's still often puzzling to figure out exactly what they mean, especially if you're running something like Google Analytics too. As you say, shouldn't they all add up?

The answer is, no, they don't all add up. The problem is that you're measuring different things with these different programs. In your case, for example, when you say you have 10,000 page views in a typical day, that likely includes pages requested by Googlebot and other spiders that are indexing your content, pages requested by mobile devices or similar that can't get all of the elements and might simply ignore requests to include external data and users who have disabled or turned off JavaScript, which prevents the ad from showing up. In addition, some people run ad blocking software which means that they tend to get blank space where ads would otherwise be shown.

More important than all of these, though, the main reason that you're seeing about 50% of your page views having AdSense impressions, is because Google can't always fulfill a requested ad block with paid advertising.

Advertisers call this the "inventory problem" and it works like this: let's say that there are five possible advertisers who would be willing to pay for clicks off your pages. Each of them has a certain daily budget and early in the day every page your visitor sees has an advertisement included. Some get clicks (either on your site or another) and over the course of the day many end up exhausting their daily ad budget. By the late evening, Google has a problem: your pages are being served up and asking for ads, but all of the five advertisers have run out of money to pay for the potential clicks.

There's also a geotargeting issue: if your readers are coming from geographic regions where advertisers aren't targeting (for example, Eastern Asia) then there won't be any ads to include even though there might be available ad inventory for the same page view if they were from a more [advertiser] popular geographic region (Western Europe).

Does Google show those ads anyway, knowing that it'll violate the requested budget limits of those AdWords advertisers? Or does it replace those ads with public service ads (PSAs) that are free, or does it simply leave the ad block blank?

I'll also quote from the AdSense help too, though I don't think it really clarifies much:

"A page impression is generated every time a user views a page displaying Google ads. We will count one page impression regardless of the number of ads displayed on that page. For example, if you have a page displaying three ad units and it is viewed twice, you will generate two page impressions and six ad unit impressions."

You are seeing surprisingly low figures, I will admit, with less than 50% page impressions against your traffic, but I'd really look closely at what categories you're in, what time of day you're analyzing, and where your readers come from.

Good luck.