Let the random variable \( X \) follow a Poisson distribution. If \( P(X=3)=\frac{2}{5}P(X=2) \), then \( P(X=1) \) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{6}{5}e^{-\displaystyle\frac{6}{5}} \)
For a Poisson distribution with mean \( \lambda \),
\( \displaystyle P(X=r)=\frac{e^{-\lambda}\lambda^{r}}{r!} \)
The ratio of two successive probabilities removes \( e^{-\lambda} \):
\( \displaystyle \frac{P(X=3)}{P(X=2)}=\frac{\lambda^3/3!}{\lambda^2/2!}=\frac{\lambda}{3} \)
\( \displaystyle \frac{\lambda}{3}=\frac{2}{5} \)
\( \displaystyle \lambda=\frac{6}{5} \)
Then
\( \displaystyle P(X=1)=\lambda e^{-\lambda}=\frac{6}{5}e^{-\frac{6}{5}} \)