The variable X takes two values x1 and x2 with frequencies f1 and f2 respectively. If σ denotes the standard deviation of X, then σ2=

A
$\frac{f_{1}^{2} x_{1}^{2} + f_{2}^{2} x_{2}^{2}}{f_{1} + f_{2}} - {(\frac{f_{1} x_{1} + f_{2} x_{2}}{f_{1} + f_{2}})}^{2}$
B
$\frac{f_{1} f_{2}}{{(f_{1} + f_{2})}^{2}} {(x_{1} - x_{2})}^{2}$
C
$\frac{f_{1} f_{2}}{{(f_{1} - f_{2})}^{2}} {(x_{1} - x_{2})}^{2}$
D
none of these

Solution:

We have,

$σ^{2} = \frac{f_{1} x_{1}^{2} + f_{2} x_{2}^{2}}{f_{1} + f_{2}} - {(\frac{f_{1} x_{1} + f_{2} x_{2}}{f_{1} + f_{2}})}^{2}$

$\begin{array}{l} \Rightarrow σ^{2} = \frac{1}{{(f_{1} + f_{2})}^{2}} \{\begin{array}{c} f_{1}^{2} x_{1}^{2} + f_{2}^{2} x_{2}^{2} + f_{1} f_{2} (x_{1}^{2} + x_{2}^{2}) - f_{1}^{2} x_{1}^{2} \\ - f_{2}^{2} x_{2}^{2} - 2 f_{1} f_{2} x_{1} x_{2} \end{array}\} \\ \Rightarrow σ^{2} = \frac{f_{1} f_{2}}{{(f_{1} + f_{2})}^{2}} {(x_{1} - x_{2})}^{2} \end{array}$

The variable $X$ takes two values $x_{1}$ and $x_{2}$ with frequencies $f_{1}$ and $f_{2}$ respectively. If $σ$ denotes the standard deviation of $X,$ then $σ^{2} =$

Solution:

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