"Let X = {x $\in$ N : 1 $\le$ x $\le$ 17} and Y = {ax + b: x $\in$ X and a, b $\in$ R, a 0}. If mean and variance of elements of Y are 17 and 216 respectively then a + b is equal to :"

Question

"Let X = {x $\in$ N : 1 $\le$ x $\le$ 17} and Y = {ax + b: x $\in$ X and a, b $\in$ R, a > 0}. If mean and variance of elements of Y are 17 and 216 respectively then a + b is equal to :"

Accepted Answer

"Mean of X = ${{\sum\limits_{x = 1}^{17} x } \over {17}}$ = ${{17 \times 18} \over {17 \times 2}}$ = 9

Mean of Y = ${{\sum\limits_{x = 1}^{17} {\left( {ax + b} \right)} } \over {17}}$ = 17

$\Rightarrow$ $a{{\sum\limits_{x = 1}^{17} x } \over {17}} + b$ = 17

$\Rightarrow$ 9a + b = 17 ....(1)

Given Var(Y) = 216

$\Rightarrow$ $${{\sum\limits_{x = 1}^{17} {{{\left( {ax + b} \right)}^2}} } \over {17}} - {\left( {17} \right)^2}$$ = 216

$\Rightarrow$ ${\sum\limits_{x = 1}^{17} {{{\left( {ax + b} \right)}^2}} }$ = 8585

$\Rightarrow$ (a + b)² + (2a + b)² +....+ (17a + b)² = 8585

$\Rightarrow$ 105a² + b² + 18ab = 505 ....(2)

From equation (1) & (2)

a = 3 & b = -10

$\therefore$ a + b = –7"

Let X = {x $\in$ N : 1 $\le$ x $\le$ 17} and
Y = {ax + b: x $\in$ X and a, b $\in$ R, a > 0}. If mean
and variance of elements of Y are 17 and 216
respectively then a + b is equal to :

Solution

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Let X = {x $\in$ N : 1 $\le$ x $\le$ 17} and Y = {ax + b: x $\in$ X and a, b $\in$ R, a > 0}. If mean and variance of elements of Y are 17 and 216 respectively then a + b is equal to :

Solution

About this question

More Statistics questions

Drill 25 more like these. Every day. Free.

Let X = {x $\in$ N : 1 $\le$ x $\le$ 17} and
Y = {ax + b: x $\in$ X and a, b $\in$ R, a > 0}. If mean
and variance of elements of Y are 17 and 216
respectively then a + b is equal to :