While it may be fairly argued that stifling an educated student's entrepreneurial spirit should not be the function of the immigration system - it is possible, under some circumstances, to establish or invest in a business - but the rules differe depending on whether you have OPT (optional practical training) or not.
If You Have No OPT
While you cannot work, there is no prohibition from investing or establishing a new business. But logistically, since you cannot even "volunteer" for the company you're investing in - it can create a lot of procedural hurdles to do so. Doing a business plan for later on when OPT is obtained, however, is perfectly permissible.
If you're going to start a business - make sure to incorporate or form an LLC. The reason is that a sole proprietorship will require you to engage in business, as opposed to a separate corporate entity. You must actually create a separate organization so that you're not the only one running the show.
An F-1 student can receive dividends so long as taxes are filed since this is passive investment and not active employment. Moreover, a company that is generating business can later be used, under narrow circumstances, to petition for the F-1 student in H-1B status.
If You Have OPT
Now you can work for your own company that you have established!