Let X = xN: 1 x17 and Y=ax+b: x X and a,bR, a0.

Question

Let $X = \{x\in\mathbb{N}: 1\le x\le17\}$ and $Y=\{ax+b: x\in X 	ext{ and } a,b\in\mathbb{R}, a>0\}$. 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

Solutions · 1