x   P(x)    x*P(x)
        1    b       b
        2    a      2a
        3    a      3a 
        4    b      4b
       ----------------- 
    Totals:  1      E(X)

The totals of the individual probabilities must be 1. 
The totals of the individual expectations equals the
total expectation E(X).


Therefore
summing the middle column:

b + a + a + b = 1
      2b + 2a = 1
     2(b + a) = 1
        b + a = 

Summing the right-most column:

E(X) = b + 2a + 3a + 4b

E(X) = 5b + 5a

E(X) = 5(b + a)

Substituting  for b + a,

E(X) = 5() =  = 2.5

Edwin