A fair die is rolled six times. What are the expected value and the standard deviation of the smallest number rolled?
Let $X_i, \forall i=1,...,6$ be the outcomes of the roll and $H$ the highest outcome that I obtain. So I have to find the distribution of $Z=\operatorname{min}(X_i,H)$. I wrote:
$\mathbb{P}(Z)=\mathbb{P}[(Z\cap H=1)\cup ...\cup (Z\cap H=6)]=\mathbb{P}(Z\cap H=1)+...+\mathbb{P}(Z\cap H=6)=\sum_{j=1}^{6}\mathbb{P}(H=j)\mathbb{P}(Z|H=j)$
Now the problem: how do I have to calculate $\mathbb{P}(Z=z)=\sum_{j=1}^{6}\mathbb{P}(H=j)\mathbb{P}[\operatorname{min}(X_i,H=j)=z|H=j]$?
Thanks in advance for any help?