Q62 / 200 · AP EAMCET 2002 · Mathematics

If the mean of a poisson distribution is $\frac{1}{2}$, then the ratio of $P(X=3)$ to $P(X=2)$ is

  • A
    $1: 2$
  • B
    $1: 4$
  • Correct answer
    C
    $1: 6$
  • D
    $1: 8$

Solution

Given that $\lambda=\frac{1}{2}$ Now, $P(X=n)=\frac{\lambda^n}{n !} e^\lambda$ $$ \begin{aligned} & \therefore \quad P(X=3)=\frac{\left(\frac{1}{2}\right)^3}{3 !} e^{1 / 2} \\ & \text { and } P(X=2)=\frac{\left(\frac{1}{2}\right)^2}{2 !} e^{1 / 2} \\ & \therefore \quad \frac{P(X=3)}{P(X=2)}=\frac{\frac{\left(\frac{1}{2}\right)^3 e^{1 / 2}}{3 !}}{\frac{\left(\frac{1}{2}\right)^2 e^{1 / 2}}{2 !}}=\frac{\left(\frac{1}{2}\right)^3 e^{1 / 2} 2 !}{\left(\frac{1}{2}\right)^2 e^{1 / 2} 3 !}=\frac{\frac{1}{2}}{3}=\frac{1}{6} \\ & \end{aligned} $$

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

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