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,

पर्याय

• 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)

$$\text{ Var }(X)=\frac{\sum _{i=1}^n(x_i-{\overline{X}}^{}{)}^2}{n}\text{ where Mean }\left(X\right)=\frac{\sum _{i=1}^n{x}_i}{n}$$

$$\text{ Var }(Y)=\frac{{\sum }_{i=1}^n(y_i-Y{)}^2}{n}\text{ and }Y=\frac{\sum _{i=1}^n{y}_i}{n}$$

$$\text{ We have },$$

$$y_i=a{x}_i+b$$

$$Y=\frac{{\sum }_{i=1}^n{y}_i}{n}$$

$$=\frac{{\sum }_{i=1}^{n}a{x}_i+b}{n}$$

$$=\frac{a{\sum }_{i=1}^n{x}_i}{n}+\frac{nb}{n}$$

$$=aX+b$$

$$\text{ Var }(Y)=\frac{\sum _{i=1}^n{(y_i-Y)}^2}{n}$$

$$=\frac{\sum _{i=1}^n{\{a{x}_i+b-(aX+b)\}}^2}{n}$$

$$=\frac{\sum _{i=1}^n(a{x}_i-aX{)}^2}{n}$$

$$=a^2\frac{\sum _{i=1}^n(x_i-X{)}^2}{n}$$

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