Q70 / 120 · VITEEE 2008 · Mathematics

A random variable X follows binomial distribution with mean $\alpha$ and variance $\beta$. Then

  • A
    $0 < \alpha < \beta$
  • Correct answer
    B
    $0 < \beta < \alpha$
  • C
    $\alpha < 0 < \beta$
  • D
    $\beta < 0 < \alpha$

Solution

Mean, $\mathrm{np}=\alpha$; and variance, $\mathrm{npq}=\beta$ where $\mathrm{n}$ $=$ number of trials and $\mathrm{p}+\mathrm{q}=1$ So, $\frac{\mathrm{npq}}{\mathrm{np}}=\frac{\beta}{\alpha} \Rightarrow \mathrm{q}=\frac{\beta}{\alpha} \Rightarrow(1-\mathrm{p})=\frac{\beta}{\alpha}$ $\because 0 < \mathrm{p} < 1$ $\therefore-1 < -\mathrm{p} < 0 \Rightarrow 0 < 1-\mathrm{p} < 1$ $\Rightarrow 0 < \frac{\beta}{\alpha} < 1 \Rightarrow 0 < \beta < \alpha$

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

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