Question

Let x₁, x₂, ..., x_n be values taken by a variable X and y₁, y₂, ..., y_n be the values taken by a variable Y such that y_i = ax_i + b, i = 1, 2,..., n. Then,

विकल्प

• Var (Y) = a² Var (X)

• Var (X) = a² Var (Y)

•  Var (X) = Var (X) + b

• none of these

   

Answer 1

 Var (Y) = a² Var (X)

\[\text{ Var } (X) = \frac{\sum^n_{i = 1} ( x_i - \bar{X}^{} )^2}{n} \text{ where Mean } \left( X \right) = \frac{\sum^n_{i = 1} x_i}{n}\]

\[\text{ Var } (Y) = \frac{\sum\nolimits_{i = 1}^n ( y_i - Y )^2}{n} \text{ and } Y =  \frac{\sum^n_{i = 1} y_i}{n}\]

\[\text{ We have } , \]

\[ y_i = a x_i + b\]

\[Y = \frac{\sum\nolimits_{i = 1}^n y_i}{n}\]

\[ = \frac{\sum\nolimits_{i = 1}^n a x_i + b}{n}\]

\[ = \frac{a \sum\nolimits_{i = 1}^n x_i}{n} + \frac{nb}{n} \]

\[ = aX + b\]

\[\text{ Var } (Y) = \frac{\sum^n_{i = 1} \left( y_i - Y \right)^2}{n}\]

\[ = \frac{\sum^n_{i = 1} \left\{ a x_i + b - \left( aX + b \right) \right\}^2}{n}\]

\[ = \frac{\sum^n_{i = 1} (a x_i - aX )^2}{n}\]

\[ = a^2 \frac{\sum^n_{i = 1} ( x_i - X )^2}{n}\]

\[ = a^2 \text{ Var } (X)\]