We have to first compute the mean of the distribution in order to calculate the variance.

Mean=Sum(X.P(X=x))=(1*0.1)+(2*0.2)+(3*0.1)+(4*0.3)+(5*0.2)+(6*0.1)

=0.1+0.4+0.3+1.2+1+0.6=3.6

Variance=Sum(P(X)*(X-Mean)²)=0.1*(1-3.6)²+0.2*(2-3.6)²+0.1*(3-3.6)²+0.3*(4-3.6)²+0.2*(5-3.6)²+0.1*(6-3.6)²=2.24