- Home
- Standard 11
- Mathematics
Let the observations $\mathrm{x}_{\mathrm{i}}(1 \leq \mathrm{i} \leq 10)$ satisfy the equations, $\sum\limits_{i=1}^{10}\left(x_{i}-5\right)=10$ and $\sum\limits_{i=1}^{10}\left(x_{i}-5\right)^{2}=40$ If $\mu$ and $\lambda$ are the mean and the variance of the observations, $\mathrm{x}_{1}-3, \mathrm{x}_{2}-3, \ldots ., \mathrm{x}_{10}-3,$ then the ordered pair $(\mu, \lambda)$ is equal to :
$(6, 6)$
$(3, 6)$
$(6, 3)$
$(3, 3)$
Solution
$\sum_{i=1}^{10}\left(x_{i}-5\right)=10$
$\Rightarrow$ Mean of observation $\mathrm{x}_{\mathrm{i}}-5=\frac{1}{10} \sum_{\mathrm{i}=1}^{3}\left(\mathrm{x}_{\mathrm{i}}-5\right)=1$
$\Rightarrow \mu=$ mean of observation $\left(\mathrm{x}_{\mathrm{i}}-3\right)$
$=\left(\text { mean of observation }\left(\mathrm{x}_{\mathrm{i}}-5\right)\right)+2$
$=1+2=3$
Variance of observation
$\mathrm{x}_{\mathrm{i}}-5=\frac{1}{10} \sum_{\mathrm{i}=1}^{10}\left(\mathrm{x}_{\mathrm{i}}-5\right)^{2}-\left(\mathrm{Mean} \text { of }\left(\mathrm{x}_{\mathrm{i}}-5\right)\right)^{2}=3$
$\Rightarrow \quad \lambda=$ variance of observation $\left(\mathrm{x}_{\mathrm{i}}-3\right)$
$=$ variance of observation $\left(\mathrm{x}_{\mathrm{i}}-5\right)=3$ $\therefore \quad(\mu, \lambda)=(3,3)$
Similar Questions
Find the mean and variance for the following frequency distribution.
| Classes | $0-10$ | $10-20$ | $20-30$ | $30-40$ | $40-50$ |
| Frequencies | $5$ | $8$ | $15$ | $16$ | $6$ |
