If a random variable X follows a Poisson dist | vedclass

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Question

(A) For a Poisson distribution,the probability mass function is given by $P(X=r) = \frac{\lambda^r e^{-\lambda}}{r!}$,where $\lambda$ is the parameter

Vedclass · Accepted Answer

$\frac{4}{81} e^{-\frac{2}{3}}$

Vedclass · Answer

(A) For a Poisson distribution,the probability mass function is given by $P(X=r) = \frac{\lambda^r e^{-\lambda}}{r!}$,where $\lambda$ is the parameter of the distribution. Given that $P(X=1) = 3P(X=2)$. Substituting the formula: $\frac{\lambda^1 e^{-\lambda}}{1!} = 3 	imes \frac{\lambda^2 e^{-\lambda}}{2!}$ $\lambda = 3 	imes \frac{\lambda^2}{2}$ Since $\lambda 
eq 0$,we divide by $\lambda$: $1 = \frac{3\lambda}{2} \implies \lambda = \frac{2}{3}$. Now,we calculate $P(X=3)$: $P(X=3) = \frac{\lambda^3 e^{-\lambda}}{3!} = \frac{(\frac{2}{3})^3 e^{-\frac{2}{3}}}{6}$ $P(X=3) = \frac{\frac{8}{27} e^{-\frac{2}{3}}}{6} = \frac{8}{27 	imes 6} e^{-\frac{2}{3}} = \frac{4}{81} e^{-\frac{2}{3}}$.

$X = x$	$0$	$1$	$2$	$3$
$P(X = x)$	$\frac{1}{10}$	$\frac{2}{10}$	$\frac{3}{10}$	$\frac{4}{10}$

If a random variable $X$ follows a Poisson distribution such that $P(X=1) = 3P(X=2)$,then $P(X=3) =$

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