The mean and variance of $7$ observations are $8$ and $16,$ respectively. If five of the observations are $2,4,10,12,14 .$ Find the remaining two observations.
Let the remaining two observations be $x$ and $y$.
The observations are $2,4,10,12,14, x , y$
Mean, $\bar{x}=\frac{2+4+10+12+14+x+y}{7}=8$
$\Rightarrow 56=42+x+y$
$\Rightarrow x+y=14$
Varaiance $ = 16 = \frac{1}{n}\sum\limits_{i = 1}^7 {{{\left( {{x_i} - \bar x} \right)}^2}} $
$16=\frac{1}{7}[(-6)^{2}+(-4)^{2}+(2)^{2}$
$+(4)^{2}+(6)^{2}+x^{2}+y^{2}-2 \times 8(x+y)+2 \times(8)^{2}]$
$16=\frac{1}{7}\left[36+16+4+16+36+x^{2}+y^{2}-16(14)+2(64)\right]$ .......[ using $(1)$ ]
$16=\frac{1}{7}\left[108+x^{2}+y^{2}-224+128\right]$
$16=\frac{1}{7}\left[12+x^{2}+y^{2}\right]$
$\Rightarrow x^{2}+y^{2}=112-12=100$
$\Rightarrow x^{2}+y^{2}=100$ ........$(2)$
From $(1),$ we obtain
$x^{2}+y^{2}+2 x y=196$ .........$(3)$
From $(2)$ and $(3),$ we obtain
$2 x y=196-100$
$\Rightarrow 2 x y=96$ .........$(4)$
Subtracting $(4)$ from $(2),$ we obtain
$x^{2}+y^{2}-2 x y=100-96$
$\Rightarrow(x-y)^{2}=4$
$\Rightarrow x-y=\pm 2$ .........$(5)$
Therefore, from $(1)$ and $(5),$ we obtain
$x=8$ and $y=6$ when $x-y=2$
$x=6$ and $y=8$ when $x-y=-2$
Thus, the remaining observations are $6$ and $8 .$
Find the mean and variance for the first $10$ multiples of $3$
If the variance of $10$ natural numbers $1,1,1, \ldots ., 1, k$ is less than $10 ,$ then the maximum possible value of $k$ is ...... .
Let $ \bar x , M$ and $\sigma^2$ be respectively the mean, mode and variance of $n$ observations $x_1 , x_2,...,x_n$ and $d_i\, = - x_i - a, i\, = 1, 2, .... , n$, where $a$ is any number.
Statement $I$: Variance of $d_1, d_2,.....d_n$ is $\sigma^2$.
Statement $II$ : Mean and mode of $d_1 , d_2, .... d_n$ are $-\bar x -a$ and $- M - a$, respectively
Consider $10$ observation $\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{10}$. such that $\sum_{i=1}^{10}\left(x_i-\alpha\right)=2$ and $\sum_{i=1}^{10}\left(x_i-\beta\right)^2=40$, where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. The $\frac{\beta}{\alpha}$ is equal to :
If the standard deviation of the numbers $ 2,3,a $ and $11$ is $3.5$ then which of the following is true ?