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Question 11 Mark

If $x_1, x_2, ..., x_n $ are n values of a variable $X$ and $y_1, y_2, ..., y_n $ are $n$ values of variable $Y$ such that $y_i = ax_i + b, i = 1, 2, ..., n,$ then write $Var(Y)$ in terms of $Var(X).$

Answer

$\overline{\text{Y}}=\frac{1}{\text{n}}\Big\{\sum\text{y}_\text{i}\Big\}=\frac{1}{\text{n}}\Big\{\sum\text{ax}_\text{i}+\text{b}\Big\}=\text{a}\overline{\text{x}}+\text{b}$
$\therefore\text{y}_\text{i}-\overline{\text{y}}=\text{ax}_\text{i}+\text{b}-\text{a}\overline{\text{x}}-\text{b}=\text{a}(\text{x}_\text{i}-\overline{\text{x}})$
$\text{Var}(\text{Y})=\frac{1}{\text{n}}\Big\{\sum(\text{y}_\text{i}-\overline{\text{y}})^2\Big\}=\frac{1}{\text{n}}\Big\{\sum\text{a}^2(\text{x}_\text{i}-\overline{\text{x}})^2\Big\}\\=\text{a}^2\Big[\frac{1}{\text{n}}\big\{(\text{x}_\text{i}-\overline{\text{x}})^2\big\}\Big]=\text{a}^2\big[\text{Var}(\text{X})\big]$

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Question 21 Mark

If the sum of the squares of deviations for 10 observations taken from their mean is 2.5, then write the value of standard deviation.

Answer

$\text{Given}\sum(\text{x}_\text{i}-\overline{\text{x}})^2=2.5$
$\text{SD}=\sqrt{\frac{2.5}{10}}=0.5$

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Question 31 Mark

If $X$ and $Y$ are two variates connected by the relation $\text{Y}=\frac{\text{aX+b}}{\text{c}}$ and $\text{Var}(\text{X})=\sigma^2,$ then write the expression for the standard deviation of $Y.$

Answer

We know that if $y_i = ax_i + b,$
then $var(Y) = a^2 [Var\ (X)]$
Here, $\text{Y}=\frac{\text{aX+b}}{\text{c}}=\frac{\text{a}}{\text{c}}\text{X}+\frac{\text{b}}{\text{c}}$
Therefore, $\text{Var}(\text{Y})=\frac{\text{a}^2}{\text{c}^2}[\text{var}(\text{X})]=\frac{\text{a}^2\sigma^2}{\text{c}^2}$.
$\text{S.D.}=\Big|\frac{\text{a}}{\text{c}}\Big|\sigma$

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Question 41 Mark

If a variable $X$ takes values $0, 1, 2,..., n$ with frequencies $^nC_0,\ ^nC_1,\ ^nC_2 , ... ,\ ^nC_n, $ then write variance $X.$

Answer

$\overline{\text{x}}=\frac{\sum\limits_{\text{i}=0}^{\text{n}}\text{x}_\text{i}\text{f}_\text{i}}{\sum\limits_{\text{i}=0}^{\text{n}}\text{f}_\text{i}}=\frac{0\times^\text{n}\text{C}_\text{o}+1\times^\text{n}\text{C}_1+...+\text{n}\times^\text{n}\text{c}_\text{n}}{^\text{n}\text{C}_\text{o}+^\text{n}\text{C}_1+.....+^\text{n}\text{c}_\text{n}}$
$\Rightarrow\overline{\text{x}}=\frac{\text{n}\times2^{\text{n}-1}}{\frac{2^\text{n}}{\text{n}+1}}$
$=\frac{\text{n}(\text{n}+1)}{2}$
$\therefore\text{Var}(\text{X})=\sigma^2$
$=\frac{1}{\text{n}}\sum\limits^{\text{n}}_{\text{i}=0}\big(\text{x}_\text{i}-\overline{\text{x}}\big)^2$
$=\frac{1}{\text{n}}[(0+1+2+....+\text{n})-\text{n}\overline{\text{x}}]^2$
$\Rightarrow\sigma^2=\frac{1}{\text{n}}\Big[\frac{\text{n}(\text{n}+1)}{2}-\frac{\text{n}\times\text{n}(\text{n}+1)}{2}\Big]^2$
$=\frac{1}{\text{n}}\Big[\frac{\text{n}(\text{n}+1)}{2}(1-\text{n})\Big]^2$
$=\frac{\text{n}^2}{4\text{n}}(\text{n}+1)^2(\text{n}-1)^2$

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Question 51 Mark

If each observation of a raw data whose standard deviation is $\sigma$ is multiplied by a, then write the S.D. of the new set of observations.

Answer

Standard deviation, $\sigma=\sqrt{\frac{\sum\limits_{\text{i}}(\text{x}_\text{i}-\overline{\text{x}})^2}{\text{n}}}$
Here, $\overline{\text{x}}$ represents the arithmetic mean.
Multiplying each $x_i$ by a:
$\overline{\text{x}}_{\text{new}}=\frac{1}{\text{n}}\sum\limits_{\text{i}}\text{a}.\text{x}_\text{i}$
$=\text{a}\times\frac{1}{\text{n}}\sum_\text{i}\text{x}_\text{i}$
$=\text{a}.\overline{\text{x}}_{\text{old}}$
New standard deviation, $\sigma=\sqrt{\frac{\sum\limits_{\text{i}}(\text{a}.\text{x}_\text{i}-\text{a}.\overline{\text{x}})^2}{\text{n}}}$
$=\sqrt{\frac{\sum\limits_{\text{i}}\text{a}^2.(\text{x}_\text{i}-\overline{\text{x}})^2}{\text{n}}}$
$=|\text{a}|\sqrt{\frac{\sum\limits_{\text{i}}(\text{x}_\text{i}-\overline{\text{x}})^2}{\text{n}}}$
$=|\text{a}|.\sigma$

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Question 61 Mark

In a series of 20 observations, 10 observations are each equal to k and each of the remaining half is equal to -k. If the standard deviation of the observations is 2, then write the value of k.

Answer

Here, n = 20, $\overline{\text{x}}=0$
$\therefore\sum(\text{x}_\text{i}-\overline{\text{x}})^2=\sum(\text{x}_\text{i})^2=20\text{k}^2$
$\text{Var}(\text{X})=\frac{1}{\text{N}}\Big\{\sum\big(\text{x}_\text{i}-\overline{\text{x}}\big)^2\Big\}=\frac{20\text{k}^2}{20}=\text{k}^2$
$\text{S.D.}(\text{X})=\sqrt{\text{k}^2}=2$
$\therefore\text{k}=\pm2$

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Question 71 Mark

Write the variance of first n natural numbers.

Answer

Variance $=\sum\frac{\text{x}^2}{\text{n}}-\Big(\sum\frac{\text{x}}{\text{n}}\Big)^2$
But $\sum\text{x}=1+2+3+....+\text{n}=\text{n}\Big(\text{n}+\frac{1}{2}\Big)$
$\sum\text{x}^2=\frac{\text{n}(\text{n}+1)(2\text{n}+1)}{6}$
Substituting these value we get
Variance = $=\frac{(\text{n}^2-1)}{12}$

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