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 _____.

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 _____.

TestHub · Accepted Answer

Given,
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,
Now mean is given by,
Mean $= \frac{8 + 16 + 120 + 80 + 56 + 80 + 6 α + 12 β}{40 + α + β} = 9$
$\Rightarrow 360 + 9 α + 9 β = 360 + 6 α + 12 β$
$\Rightarrow 3 α - 3 β = 0$
$\Rightarrow α = β . . . . . . (1)$
Now variance is given by,
$15.08 = \frac{16 + 64 + 36 α + 960 + 800 + 144 α + 784 + 1280}{40 + 2 α} - 9^{2}$
$\Rightarrow 96.08 = \frac{3904 + 180 α}{40 + 2 α}$
$\Rightarrow (40 + 2 α) (96.08) = 3904 + 180 α$
$\Rightarrow 3843.20 + (192.16) α = 3904 + 180 α$
$\Rightarrow (12.16) α = 60.80$
$\Rightarrow α = 5 = β$
Hence, $α^{2} + β^{2} - α β = 25 + 25 - 25 = 25$

Mathematics - Statistics Question with Solution | TestHub

Doubts & Discussion

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