Let $ X = { x \in \mathbb{N} : 1 \le x \le 19 } $ and for some $ a, b \in \mathbb{R} $, $ Y = { ax + b : x \in X } $. If the mean and variance of the elements of $ Y $ are $ 30 $ and $ 750 $, respectively, then the sum of all possible values of $ b $ is

$ \sum y_i = a \sum x_i + \sum b = a(1 + 2 + \cdots + 19) + 19b $

$ \sum y_i = \frac{a \cdot 19 \cdot 20}{2} + 19b $

$ \Rightarrow 30 = 10a + b \quad ...(1)$

Variance of $ X = \frac{\sum x_i^2}{19} - \left( \frac{\sum x_i}{19} \right)^2 $

$ = \frac{19 \cdot 20 \cdot 39}{19 \cdot 6} - (10)^2 = 30 $

Variance of $ Y = a^2 \times (\text{variance of } X)$

$ 750 = a^2 \times 30 \Rightarrow a^2 = 25 \Rightarrow a = \pm 5 $

if $ a = 5 \Rightarrow b = 30 - 50 = -20 $

if $ a = -5 \Rightarrow b = 30 + 50 = 80 $

sum of values of $ b = 80 - 20 = 60 $