Assuming, perhaps incorrectly, that the system in Switzerland is similar to that of Germany, you will find useful information, but not an answer here.
It isn't reasonable to expect that any entering doctoral student will have already developed a research idea to the extent that they have a solution, as that is the purpose of the dissertation research.
But in a system in which the primary work toward the degree is precisely that research and the student starts with a fairly high level of knowledge and skill, many, probably most, advisors will want you to be able to state a problem to be solved (question to be answered) and something of the methodology that you propose to attack it. That can fit on one page, provided that the methodology isn't, itself, novel.
And, of course, the problem has to "fit" within the broad outlines of the interests of the advisor.
So, if your proposal is in theoretical statistics the methodology is likely largely mathematical, while if you are applying statistics to some problem you likely know how to state the outline of the methodology succinctly, since your audience of one is probably an expert also.
But if your problem is so trivial that you can already suggest a solution, then you may need to dig deeper. The exceptional case would be if you have already done a preliminary study and intend to dig deeper. In such a case, a possible solution can be suggested.
But, focus on the broad outline of the problem and say a few words about methodology in the typical situation. One page. In naming the problem, some explanation of why it is important is also likely a good idea. And, if you are following up an earlier work, say a few words about that.
And, that one page may just be an introduction to a longer conversation with the advisor.