Option 1: This statement is correct. The Speaker of Lok Sabha must be a member of the House of the People (Lok Sabha). If the Speaker ceases to be a member of the house, he or she automatically vacates the office.
Option 2: This statement is incorrect. The Speaker cannot simply resign by writing a letter to the President of India. The Speaker can only be removed from office by a resolution passed by the House of the People (Lok Sabha). Resignation must be submitted to the Deputy Speaker.
Option 3: This statement is correct. The Speaker can be removed from office by a resolution passed by the majority of all the members of the House of the People.
Option 4: This statement is correct. When the office of the Speaker is vacant, the duties of the office shall be performed by the Deputy Speaker.
In conclusion, option 2 is the correct answer as it states an incorrect statement about the Speaker of Lok Sabha.