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જો $n$ અવલોકનો $x_1, x_2, x_3.........x_n$ ના મધ્યક $\bar x$ અને વિચરણ $\sigma ^2$ હોય, તો સાબિત કરી કે અવલોકનો $a x_{1}, a x_{2}, a x_{3}, \ldots ., a x_{n}$ ના મધ્યક અને વિચરણ અનુક્રમે $a \bar{x}$ અને $a^{2} \sigma^{2}$ છે, $(a \neq 0)$.
Solution
The given n observations are $x _{1}, x _{2} \ldots x _{ n }$
Mean $=\bar{x}$
Variance $=\sigma^{2}$
$\therefore {\sigma ^2} = \frac{1}{n}\sum\limits_{i = 1}^n {{y_1}{{\left( {{x_i} – \bar x} \right)}^2}} $ ……….$(1)$
If each observation is multiplied by a and the new observations are $y_{i},$ then
$y_{1}=a x_{i}$ i.e., $x_{i}=\frac{1}{a} y_{1}$
$\bar y = \frac{1}{n}\sum\limits_{i = 1}^n {{y_1} = \frac{1}{n}\sum\limits_{i = 1}^n {a{x_1} = \frac{a}{n}} } \sum\limits_{i = 1}^n {{x_1} = ax} $ $\left( {\bar x = \frac{1}{n}\sum\limits_{i = 1}^n {{x_1}} } \right)$
Therefore, mean of the observations, $a x_{1}, a x_{2} \ldots a x_{n},$ is $a \bar{x}$
Substituting the values of $x_{i}$ and $\bar{x}$ in $(1),$ we obtain
${\sigma ^2} = \frac{1}{n}{\sum\limits_{i = 1}^n {\left( {\frac{1}{a}{y_1} – \frac{1}{a}\bar y} \right)} ^2}$
$ \Rightarrow {a^2}{\sigma ^2} = \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{y_1} – \bar y} \right)}^2}} $
Thus, the variance of the observations, $a x_{1}, a x_{2} \ldots a x_{n},$ is $a^{2} \sigma^{2}$