is it common for babies to put finger in their mouth   

There is nothing to worry about this habit as it will gradually reduce and might go as your child turns six to seven month old, however, you can try these tips to control your baby: 1.) Feed (Breastfeed or formula feed) immediately when he/she puts hand in the mouth. 2.) Try to grab your child’s attention towards toys, rattles or other activity. 3.) You can also let them enjoy this activity as it is a good sign of growth, check if the child is hungry or change the diaper if it's wet 4.) You can distract your baby by singing a song, playing or showing toys with interesting sound/music.