Question

In a poisson frequency distribution, frequency corresponding to 3 successes in 2/3 times frequency corresponding to 4 successes. Find the mean and standard deviation. 

Answer 1

Denote a number of successes as $k$ .

Then probability for Poisson distribution is

$P(k)=\dfrac{\lambda^k\cdot e^\lambda}{k!}$

Mean and variance are both equal to $\lambda$ .

So,

mean $\bold{E}[X]=\lambda$

standard deviation $\sigma=\sqrt{\bold{Var}[X]}=\sqrt{\lambda}$ .

$P(3)=\dfrac{\lambda^3\cdot e^\lambda}{3!}$

$P(4)=\dfrac{\lambda^4\cdot e^\lambda}{4!}$

$P(3)=\dfrac{2}{3}P(4)$

$\dfrac{\lambda^3\cdot e^\lambda}{3!}=\dfrac{2}{3}\cdot\dfrac{\lambda^4\cdot e^\lambda}{4!}$

$\dfrac{\lambda^3}{3!}=\dfrac{2}{3}\cdot\dfrac{\lambda^4}{4!}$

$\dfrac{1}{3!}=\dfrac{2}{3}\cdot\dfrac{\lambda}{4!}$

$\dfrac{\lambda}{4!}=\dfrac{3}{2}\cdot\dfrac{1}{3!}$

$\lambda=\dfrac{3}{2}\cdot\dfrac{4!}{3!}$

$\lambda=\dfrac{3}{2}\cdot\dfrac{3!\cdot4}{3!}=3\cdot2=6$

Answer

Mean $\bold{E}[X]=6$

Standard deviation $\sigma=\sqrt{6}$