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GO Classes DPP | STATISTICS |VARIANCE

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A random variable $X$ has mean $\mu$ and variance $\sigma^2$.

A new variable $Y$ is defined as $Y=a X+b$, where $a$ and $b$ are constants.

It is known that $\operatorname{Var}(Y)=9 \times \operatorname{Var}(X)$.

If the mean of $Y$ is $45$ when the mean of $X$ is $15$ , find the values of $a$ and $b$.
 

  1. $a=3, b=0$
     
  2. $a=-3, b=90$
     
  3. Both A and B
     
  4. $a=9, b=-90$

1 Answer

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$\operatorname{Var}(\mathrm{Y})=\mathrm{a}^2 \operatorname{Var}(\mathrm{X})=9 \operatorname{Var}(\mathrm{X}) \Rightarrow \mathrm{a}^2=9 \Rightarrow \mathrm{a}=3$ or $\mathrm{a}=-3$.

Mean relation: $\mathrm{E}[\mathrm{Y}]=\mathrm{a} \mathrm{E}[\mathrm{X}]+\mathrm{b}$. Given $\mathrm{E}[\mathrm{Y}]=45$ and $\mathrm{E}[\mathrm{X}]=15$, so $15 \mathrm{a}+\mathrm{b}=45 \Rightarrow \mathrm{~b}=45-15 \mathrm{a}$.

If $\mathrm{a}=3$ then $\mathrm{b}=0$. If $\mathrm{a}=-3$ then $\mathrm{b}=90$.

Therefore $(\mathrm{a}, \mathrm{b})=(3,0)$ or $(-3,90)$.
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