I am using the Poission distribution to calculate probabillity for x = X.

Poisson distribution is skewed to the right when lambda is small.

Problem I have is that distribution for my x is rather skewed to the left when the expectancy.

Is there any other known distribution I can use to calculate x = X, that is skewed to the left when expectancy for x is small?

Is it is possible to use the binomial distribution where I use p as "skewness parameter" and n be 'expected value'/prob?

Really appreciate som help for this.