Question

The mean and variance of 5 observations are 3 and 2 respectively. If three of the observations are 1,3 &5. Find the value of other two observations.​

Answer 1

$\color {red}\huge\bold\star\underline\mathcal{Question:-}$ We have to find the value of other 2 observations.

$\huge\underline\blue{\sf Given:}$

--->Mean is 3

---->variance is 2 .

---> Total observations are 5 ..

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$\bold{\boxed{\huge{\boxed{\orange{\small{\boxed{\huge{\red{\bold{\:Answer}}}}}}}}}}$ :-------

We know that ,

$\large\red{\boxed{\sf </strong><strong>Mean</strong><strong> \: = \frac{sum \: of \: observation}{total \: observation}}}$

Let two other observations are = X & Y .

Than ,

$3 = \frac{1 + 3 + 5 + x + y}{5} \\ \\ 15 = 9 + x + y \\ \\ x + y = 6 - - - - (equation1)$

Now , we know That ,

$variance = \frac{∑xi {}^{2} }{5} - x ^{2} \\ \\ variance = \frac{ (1 + 9 + 25 + {x}^{2} + {y}^{2} ) }{5} - 9 \\ \\ (2 + 9) \times 5 = 35 + {x}^{2} + {y}^{2} \\ \\ {x }^{2} + {y}^{2} = ( 20) - - - equation(2)$

now, squaring both sides in equation (1) we get,

$(x + y) {}^{2} = 36 \\ \\ ( {x}^{2} + {y}^{2} + 2xy) = 36 \\ \\ 2xy = 36 - 20 = 16$

Also, we know that ,

$(x - y) {}^{2} = {x}^{2} + {y}^{2} - 2xy \\ \\ (x - y) {}^{2} =20 - 16 \\ \\ (x - y) = 2 \: or (- 2) - - - equation(3)$

so, from (1) and (3) now,

$when \: (x-y) = 2, \\ \\ x + y = 6 \\ x - y = 2 \\ \\ 2x = 8 \\ x = 4 \\ y = 2 \\ \\ \\ and \: when(x - y) = ( - 2) \\ we \: get \\ \\ x + y = 6 \\ x - y = - 2 \\ 2x = 4 \\ x = 2 \\ y = 4$

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$\huge\blue{THANKS}$

$\large\red{\boxed{\sf </strong><strong>Nice\</strong><strong>:</strong><strong>Question</strong><strong>}}$