Question

Let X∼Poisson(k−1)X∼Poisson(k−1) with Var(X)=1Var(X)=1, then find the value of the constant kk

Answer 1

We know that if $X$ follows a Poisson distribution of parameter $\lambda$ , i.e. $X\sim \textrm{Poisson}(\lambda)$ , then the variance of $X$ is $\textrm{Var}(X)=\lambda$.

If we apply it to our case we have:

$$\textrm{Var}(X)=k-1=1\Rightarrow k-1=1\Rightarrow k=1+1\Rightarrow k=2$$

Thus, the value of the constant $k$ is $k=2$ .