Question

12. In a Poisson distribution if 2P(x=1)=P(x=2), then the variance is

A. O

B. -1

C.4

D. 2​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{In a poisson distribution,}$

$\mathsf{2\,P(X=1)=P(X=2)}$

$\underline{\textsf{To find:}}$

$\textsf{The variance}$

$\underline{\textsf{Solution:}}$

$\underline{\mathsf{Concept:}}$

$\mathsf{The\;probability\;mass\;function\;of\;poisson\;distributiion\;is}$

$\mathsf{P(X=x)=\dfrac{e^{-\lambda}\,{\lambda}^x}{x!}\;\;x=0,1,2,3........}$

$\mathsf{Consider,}$

$\mathsf{2\,P(X=1)=P(X=2)}$

$\implies\mathsf{2\left(\dfrac{e^{-\lambda}\,{\lambda}^1}{1!}\right)=\dfrac{e^{-\lambda}\,{\lambda}^2}{2!}}$

$\implies\mathsf{2=\dfrac{{\lambda}}{2}}$

$\implies\mathsf{\lambda=2{\times}2}$

$\implies\boxed{\mathsf{\lambda=4}}$

$\therefore\textsf{Variance is 4}$

$\underline{\textsf{Answer:}}$

$\mathsf{Option (C) is correct}$

$\underrline{\textsf{Find more:}}$

If the mean of poisson variable x is one what is p(x =takes the value at least 1)

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