Let \(X\sim\text{Poisson}(\lambda)\) and define a new random variable \(Y\) that is a zero-truncated version of \(X\). That is, we start with \(\text{Range}(X)=\{0,\,1,\,2,\,3,\,...\}\), and then we rig it so that \(\text{Range}(Y)=\{1,\,2,\,3,\,...\}\). To do this, we define the PMF of the new random variable \(Y\) by

\[ P(Y=k) = P(X=k\mid X > 0),\quad k=1,\,2,\,3,\,4,\,... \]

1. Show that the new PMF sums to one;
2. Compute the mean of \(Y\);
3. Compute the variance of \(Y\).