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Let X1, X2, ..., Xn Be Values Taken by a Variable X and Y1, Y2, ..., Yn Be the Values Taken by a Variable Y Such Yi = Axi + B, I = 1, 2,..., N. Then, (A) Var (Y) = A2 Var (X) (B) Var (X) = A2 Var (Y) - Mathematics

MCQ

Let x1x2, ..., xn be values taken by a variable X and y1y2, ..., yn be the values taken by a variable Y such that yi = axi + bi = 1, 2,..., n. Then,

Options

  • Var (Y) = a2 Var (X)

  • Var (X) = a2 Var (Y)

  •  Var (X) = Var (X) + b

  • none of these

     
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Solution

 Var (Y) = a2 Var (X)

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

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

\[\text{ We have } , \]

\[ y_i = a x_i + b\]

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

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

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

\[ = aX + b\]

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

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

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

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

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

 

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 32 Statistics
Q 11 | Page 51
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