Deriving a constant in a probability mass function

Question

Let $Y_1, \ldots , Y_n$ be a random sample from a zero-truncated Poisson distribution with probability mass function:

$$p_Y (y\mid λ) = k\frac{λ^ye^{-λ}}{y!},$$ $k > 0, λ > 0, y = 1, 2, \ldots ,$ where $k$ is an unknown constant.

I know that the sum of a probability mass function is $1$, but I do not know how to use this information to find $k$.

Answer 1

Since $p$ is a probability mass function $$1 =\sum _{y=1}^\infty k\frac{λ^ye^{-\lambda}}{y!} =ke^{-\lambda}\sum _{y=1}^\infty\frac{λ^y}{y!} =ke^{-\lambda}(e^{\lambda}-1). $$

Answer 2

If you don't know that $\displaystyle \sum_{y=0}^\infty \frac{\lambda^y}{y!} = e^\lambda,$ then the form of the probability mass function of the Poisson distribution will tell you that.

It follows that $\displaystyle\sum_{y=1}^\infty \frac{\lambda^y}{y!} = e^\lambda - 1.$