If the variance of x_1, x_2, ldots, x_n is si | vedclass

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(D) The variance of a set of observations $x_i$ is defined as $\sigma_x^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2$,where $\bar{x}$ is the mean.

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$\lambda^2 \cdot \sigma_x^2$

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(D) The variance of a set of observations $x_i$ is defined as $\sigma_x^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2$,where $\bar{x}$ is the mean. When each observation is multiplied by a constant $\lambda$,the new mean becomes $\bar{x}' = \lambda \bar{x}$. The new variance is $\sigma_{new}^2 = \frac{1}{n} \sum_{i=1}^{n} (\lambda x_i - \lambda \bar{x})^2$. $\sigma_{new}^2 = \frac{1}{n} \sum_{i=1}^{n} \lambda^2 (x_i - \bar{x})^2$. $\sigma_{new}^2 = \lambda^2 \left[ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \right] = \lambda^2 \sigma_x^2$. Therefore,the new variance is $\lambda^2 \sigma_x^2$.

If the variance of $x_1, x_2, \ldots, x_n$ is $\sigma_x^2$,then the variance of $\lambda x_1, \lambda x_2, \ldots, \lambda x_n$ (where $\lambda \neq 0$) is:

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