Question

In a certain Poisson frequency distribution the frequency corresponding to 3 successes is one - third the frequency corresponding to 4 successes. Find its mean and standard deviation.

Answer 1

The Poisson probability is

$$P(X=k)={{e^{-\lambda}*\lambda^k} \over {k!}}$$

The frequency corresponding to 3 successes is

$$P(X=3)={{e^{-\lambda}*\lambda^3} \over {3!}}$$

The frequency corresponding to 4 successes is

$$P(X=4)={{e^{-\lambda}*\lambda^4} \over {4!}}$$

Given that the frequency corresponding to 3 successes is one-third the frequency corresponding to 4 successes

$${{P(X=3)} \over {P(X=4)}}={1 \over 3}$$

$${{e^{-\lambda}*\lambda^3*4!} \over {e^{-\lambda}*\lambda^4*3!}}={1 \over 3}$$

$$\lambda =12$$

The mean of the distribution is

$$\mu=\lambda=12$$

The variance of the distribution is

$$Var(X)=\sigma^2=\lambda=12$$

The standard deviation is

$$\sigma=\sqrt{12}=2\sqrt{3}$$