Why is the direction of cathode rays in discharge tube independent of position of anode [Still confused]?

I'd say it's wrong. In fact, cathode rays (electrons) have to be carefully focused using both electrostatic and magnetic fields. The anode attracts the electrons statically. The electrons wouldn't necessarily even make it to the anode without the anode's large potential difference with the cathode. Electrons go where the field tells them to go, and the fierld is electrostatically set up by the anode!