If X has a poisson distribution and P(X=0)=1/2,what is E(x)?

Question

If X has a poisson distribution and P(X=0)=1/2,what is E(x)?

Answer 1

The Poisson distribution can be derived from the binomial distribution.  The Poisson is nothing more than the limiting case of the Binomial where n is large and p is small.

A good way to identify when you need to use the Poisson distribution is when the problem requires you to use a rate.  This is not always true, but more often than not remembering this will help you to identify a Poisson model.

X has the Poisson distribution with parameter λ

In general you have:

X ~ Poisson( λ )

P(X = x) = ( λ )^x * exp( -λ ) / x! for x = 0, 1, 2, 3, 4, ...

P(X = x) = 0 otherwise

the mean of the Poisson distribution is the parameter, λ

the variance of the Poisson distribution is the parameter λ

P(X = 0)

= λ ^ 0 * exp(-λ) / 0!

= exp(-λ) = 1/2

log(exp(-λ)) = log(1/2)

-λ = log(1/2)

λ = - log(1/2)

λ = 0.6931472

the mean is

λ = 0.6931472