You asked: "So question is if a brokerage does that kind of fraud, will the investor be covered if he/she can show the proof that the security was purchased per the brokerage statement?"
The answer is no, the SIPC would only become involved if the firm becomes insolvent. The SIPC protects against the failure of an investment firm, not fraud by an employee or employees of the firm. That is specifically stated on their website including the quote: "SIPC was not chartered by Congress to combat fraud." You can confirm the details through the links on the page I linked, including to the Securities Investor Protection Act of 1970
The fraud you described would likely to be covered under the firm’s errors and omissions insurance, commonly known as E&O insurance. So to protect yourself you could attempt to verify the firm’s E&O policy is in force and it’s terms.
If it’s not covered by that then fraud is illegal and you can simply sue.
You could also file a complaint through the SEC or FINRA.
You also asked: "So question is how does a common investor make sure that securities have been really purchased?"
You really should be able to rely on the statements the brokerage sends. The Madoff ruling that the securities weren't actually purchased seems to be a cop out because the Madoff fraud was so large that it overwhelmed the SIPCs ability to cover it. But should isn't something you can rely on so the answer to the question is if you don't trust the firm you are investing with you would have to obsessively verify the transactions on each statement. Here is a detailed article on the statements from the Madoff scandal on how the fraud could have been caught. One of the tips from that article is:
In the past, there have been cases where brokers at large Wall Street
firms diverted the customers actual statements to their own post
office boxes and issued fraudulent statements akin to what Madoff was
doing. To safeguard against that possibility, investors can directly
call the back office of their investment firm once a year (not the
local branch phone number) and ask to have a statement mailed directly
to them while also checking that the address of record for the
statements is their actual address.
Also, Here are some tips from FINRA on checking out investment firms.
The short answer is it would be difficult to truly verify that the securities on your statement were actually purchased. You would need to trace the actual trades through any clearing firms, verify the amount and price of the transaction, and other details. Luckily this type of fraud is very difficult to pull off without giving away some signs that there is fraud.
Another way to verify your securities exist in your account would be to transfer them in kind to another brokerage firm that is independent of the one you purchased them through. If they were fraudulent trades they wouldn't be able to transfer the securities.