Nothing to do with PR. PR will only go up by getting backlinks. The PR of the page that the backlink is on will pass on some "PR juice" to the destination of the link. The higher the PR of the page the link is on.....the more "juice" will be passed.
These two are not related at all. Pagerank is Google's standard of measurement on the relevance of a site in the web. Alexa rank looks at the traffic rank of your site, and the lower the number is, the better.