Question

If x₁, x₂, ..., x_n are n values of a variable X and y₁, y₂, ..., y_n are n values of variable Y such that y_i = ax_i + b; i = 1, 2, ..., n, then write Var(Y) in terms of Var(X).

 

Answer 1

\[\text{ Var } \left( X \right) = \frac{\sum \left( x_i - \bar{X} \right)^2}{n}\]
\[\text { Var } \left( Y \right) = \frac{\sum \left( y_i - \bar{Y} \right)^2}{n}\]

We have: \[y_i = a x_i + b\]
\[ \bar{y} = \frac{\sum y_i}{n} = \frac{a\sum x_i + nb}{n} = a \bar{X} + b\]

\[\text{ Var } \left( Y \right) = \frac{\sum\nolimits_{} \left( a x_i + b - a \bar{X} - b \right)^2}{n} = \frac{\sum \left( a x_i - a \bar{X} \right)^2}{n} = a^2 \frac{\sum \left( x_i - \bar{X} \right)^2}{n} = a^2 \text{ Var } \left( X \right)\]