Let (X) be a random variable with range ({1,2, ldots}) and probability distribution [P(X=i)=left(1-frac{1}{n^{2}} ight) frac{1}{n^{2(i-1)}} ; i=1,2, ldots] Determine the (z)-transform of (X) and

Let \(X\) be a random variable with range \(\{1,2, \ldots\}\) and probability distribution

\[P(X=i)=\left(1-\frac{1}{n^{2}}\right) \frac{1}{n^{2(i-1)}} ; i=1,2, \ldots\]

Determine the \(z\)-transform of \(X\) and by means of it \(E(X), E\left(X^{2}\right)\), and \(\operatorname{Var}(X)\).