The arithmetic mean of the data 0,1,2, , n with frequencies 1, ⁴ C ₁ ⁴ C ₂, . ., ^ C _ a is

Step-by-step solution

Since, Mean = f _ i x _ i f _ i where x _ i are observations with frequencies f _ i , i =1,2, The required mean is given by aligned X &= 0.1+1 . ^ n C ₁+2 ^ n C ₂+ + n . ^ n C _ n 1+ ^ n C ₁+ ^ n C ₂+ + ^ n C _ n =& _ r =0 ^ n r ^ n C _ r _ r =0 ^ n ^ n C _ r = _ r =1 ^ n r n r ^ n -1 C _ r -1 _ r =0 ^ n ^ n C _ r =& n ^ n -1 C _ r -1 _ r =0 ^ n ^ n C _ r = n .2^ n -1 2^ n = n 2 aligned