I am going to apply to Statistics PhD but I am not deeply aware of the opportunities of research because of my different background. Is there any source of past research proposals or mock ones (maybe already provided by other programs) to get an idea of what it should be like? It would be nice to understand clearly how to structure it and how deep into the research it should actually go.
Awarded NSF proposals in the US can be requested from NSF via the Freedom of Information Act (FOIA). They give the authors a chance to redact non-public information like salaries or corporate secrets, but NSF gets a chance to edit it back in. Salaries are weird because lots of us work for state universities and our salaries are public already. The requestor’s name is not anonymous. My organization has been subject to many of these, every time we win a new big computer. Other agencies are subject to similar rules but may have different responses based on their missions (DARPA, NNSA, etc). Doing such a request is tedious and exacting, and you’re probably better off asking colleagues you’re not likely to compete against to look at one of their successful proposals.
A research proposal for a phd program? I would try asking an existing phd student what they did. If your currently doing a master's then your supervisor should be able to help. You could even contact someone at the University your applying for and see what they are looking for (this has the benefit of also getting you slightly more noticed).