What is the expectation of the first time dice-rolls Z and Z+1 are similar?

Question

We roll a dice. Let $Z=$ the first time the rolls $Z$ and $Z+1$ are similar.

What is $E[Z]$ ?

What is $E[Z^2]$ ?

The problem sounds unnatural to me because i'm asked about a roll $(Z+1)$ that hasn't happened by the time $Z$.

I tried to change the problem to: rolling $2$ dices and calculating the Expectaion of the time that they are similar:

$E[Z]=\sum i(\frac{5}{6})^{i-1}\frac{1}{6}=6$

Which is wrong.

Any direction?

Thanks in advance

Answer 1

What you have here is basically a geometric distribution.

http://en.wikipedia.org/wiki/Geometric_distribution

Your variable $Z$ is geometrically distributed: $Z \sim Geo \left(\frac{1}{6}\right)$. You can prove this by showing that $\Pr (Z = k) = \frac{1}{6} \left( \frac{5}{6}\right)^{k-1}$.

Therefore we can use known facts about the geometric distribution, so: $$ \mathbb{E}[Z] = 6. $$ $$ \mathbb{E}[Z^2] = \frac{2-\frac{1}{6}}{\frac{1}{6^2}} = 66. $$