Question

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

Answer 1

$\textbf{Given:}$

$\text{Mean of poisson variable X is 1}$

$\textbf{To find:}$

$P(X{\geq}1)$

$\textbf{Solution:}$

$\textbf{We know that,}$

$\textbf{The probability mass function of poison distribution with mean \lambda is}$

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

$\text{Mean of poisson variable X is 1}$

$\implies\bf\lambda=1$

$\text{Now,}$

$P(X{\geq}1)$

$=1-P(X\,<\,1)$

$=1-P(X=0)$

$=1-\dfrac{e^{-1}{1}^0}{0!}$

$=1-\dfrac{e^{-1}}{1}$

$=1-e^{-1}$

$=1-e^{-1}$

$\textbf{Answer:}$

$\bf\,P(X{\geq}1)=1-e^{-1}$