If the mean and variance of the frequency distribution, are 9 and 15.08 respectively, then the value of is ________ . [2023]

Question

If the mean and variance of the frequency distribution

$x_{i}$	2	4	6	8	10	12	14	16
$f_{i}$	4	4	$α$	15	8	$β$	4	5

are 9 and 15.08 respectively, then the value of $α^{2} + β^{2} - α β$ is ________ . [2023]

Smart Achievers · Accepted Answer

(25)

$Mean (\bar{x}) = \frac{\sum f_{i} x_{i}}{\sum f_{i}} \Rightarrow \frac{360 + 6 α + 12 β}{40 + α + β} = 9$

$\Rightarrow 360 + 6 α + 12 β = 360 + 9 α + 9 β$

$\Rightarrow 3 α = 3 β \Rightarrow α = β$ ... $(i)$

$Also, variance σ^{2} = \frac{\sum f_{i} x_{i}^{2}}{\sum f_{i}} - {(\frac{\sum f_{i} x_{i}}{\sum f_{i}})}^{2}$

$\Rightarrow \frac{3904 + 36 α + 144 β}{40 + α + β} - {(\bar{x})}^{2} = 15.08$

$\Rightarrow \frac{3904 + 180 α}{40 + 2 α} - {(9)}^{2} = 15.08$ $\Rightarrow α = 5$

From (i), $β = 5$

$∴$ $α^{2} + β^{2} - α β = 25 + 25 - 25 = 25$

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