If you have Uniform$(a, b)$ and Discrete Uniform$(a, b)$, why is the EV of the minimum in $k$ RVs sampled from these $2$ different, and why is it smaller in discrete uniform.
For example, if we have a Uniform$(1, 6)$, the EV of the minimum of $4$ rolls is $2$. I got this by splitting the length $5$ number line into $5$ parts ($4$ split points) and the expected length is $1$, meaning the minimum is $1+1=2$.
Now if we ask what the minimum of $4$ die rolls, I ran a simulation and it provides with $1.754$. Why are these different and why is the discrete case smaller.
While an analytical proof is not super complex, I am more looking for the intuition behind this. Thanks!