Q79 / 160 · AP EAMCET 2023 (16 May Shift 2) · Mathematics

The value of the constant $\mathrm{c}$, so that $\mathrm{P}(\mathrm{x})=\mathrm{c}\left(\frac{2}{3}\right)^{\mathrm{x}}$, $x=1,2,3, \ldots$ is the probability distribution function of a discrete random variable $\mathrm{X}$ is

  • A
    $\frac{1}{3}$
  • Correct answer
    B
    $\frac{1}{2}$
  • C
    $1$
  • D
    $0$

Solution

Given $P(X)=C\left(\frac{2}{3}\right)^X \quad X=1,2,3, \ldots$ Now, $\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2)+\mathrm{P}(\mathrm{X}=3)+\ldots=1$ $\begin{aligned} & \Rightarrow C \frac{2}{3}+C\left(\frac{2}{3}\right)^2+C\left(\frac{2}{3}\right)^3+\ldots=1 \\ & \Rightarrow C\left\{\frac{2}{3}+\left(\frac{2}{3}\right)^2+\left(\frac{2}{3}\right)^3+\ldots\right\}=1 \\ & \Rightarrow C\left\{\frac{\frac{2}{3}}{1-\frac{2}{3}}\right\}=1 \Rightarrow C=\frac{1}{2} \end{aligned}$

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

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