A fair dice is rolled until two different numbers are observed. Let X be the total number of dice is rolled. Find expected value and variance of X.
So, it can be noticed that we should have a minimum of two throws. The probability that we stop of the n throws is
$$\left(\frac{1}{6} \right)^{n-2} \left(\frac{5}{6} \right) $$
This is since after the first roll, each roll you have $\left(\frac{5}{6} \right) $ chance of getting different numbers. So, I am not sure where to go from here.
Best Answer
Guide:
We can view $X$ as $1+Y$ where $Y\sim Geo(\frac56)$.
Hence $E(X)=1+E(Y)$. I will leave the task of determining variance to you.