Q255 / 270 · NDA 2024 (Phase 1) · Mathematics

If two random variable X and Y are connected by relation $\frac{2 \mathrm{X}-3 \mathrm{Y}}{5 \mathrm{X}+4 \mathrm{Y}}=4$ and X follows Binomial distribution with parameters $n=10$ and $p=\frac{1}{2}$, then what is the variance of Y ?

  • Correct answer
    A
    $\frac{810}{361}$
  • B
    $\frac{9}{19}$
  • C
    $\frac{21}{361}$
  • D
    $\frac{121}{361}$

Solution

$\because \quad \frac{2 x-3 y}{5 x+4 y}=4$ $\begin{aligned} & \Rightarrow \quad 2 x-3 y=20 x+16 y \\ & \Rightarrow \quad 19 y=18 x \\ & \Rightarrow \quad y=\frac{18 x}{19} \\ & \operatorname{var} .(x)=n p q=n p(1-p) \\ & =10 \times \frac{1}{2} \times \frac{1}{2}=\frac{5}{2} \\ & \text { Now, } \quad \text { var. }(y)=\operatorname{var} .\left(-\frac{18}{19} x\right) \\ & \Rightarrow \quad \text { var. } \cdot(y)=\left(\frac{18}{19}\right)^2 \times \frac{5}{2}=\frac{810}{361} \end{aligned}$

Hence, the correct answer is option: 'A'.

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