Mean and variance of the wait for heads

A fair coin is flipped until the first head shows up. Let $N$ count all flips including that head. Find $\mathbb{E}[N]$ and $\operatorname{Var}(N)$.

Mean $2$, variance $2$. $N$ is geometric with $p=\tfrac12$, so $\mathbb{E}[N]=\tfrac1p=2$ and $\operatorname{Var}(N)=\tfrac{1-p}{p^2}=2$.