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Statistics — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsStatistics1 Mark
Question
If $X$ and $Y$ are two variates connected by the relation $\text{Y}=\frac{\text{aX+b}}{\text{c}}$ and $\text{Var}(\text{X})=\sigma^2,$ then write the expression for the standard deviation of $Y$.
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Answer

We know that if $y_i = ax_i + b$, then var $(Y) = a^2 [Var (X)]$
Here, $\text{Y}=\frac{\text{aX+b}}{\text{c}}=\frac{\text{a}}{\text{c}}\text{X}+\frac{\text{b}}{\text{c}}$
Therefore, $\text{Var}(\text{Y})=\frac{\text{a}^2}{\text{c}^2}[\text{var}(\text{X})]=\frac{\text{a}^2\sigma^2}{\text{c}^2}$.
$\text{S.D.}=\Big|\frac{\text{a}}{\text{c}}\Big|\sigma$

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