All simulators
Linearity of Expectation & Variance Scaling
Transform X into aX + b and watch the mean shift and the spread scale by |a|.
Day 3: Random Variables, Expectation & VarianceX ~ N(10, 4)Y = aX + b
E[Y] = aE[X] + b
10.00
1.0 × 10 + 0.0
Var(Y) = a²Var(X)
4.00
1.00 × 4
sd(Y) = |a| sd(X)
2.00
E[X], Var(X)
10.00, 4.00
Shifting by b moves the centre but not the spread; multiplying by a stretches the spread by |a| and the variance by a².