Cumulative distribution and probability mass functions. I was faced on my probability course with the following problem:
A die is rolled four times. Let $U$ be the smallest value. Calculate the probability density function and probability mass function of the random variable $U$.
How can this be solved? Now, I started solving this, but apparently I calculated the probability mass function for U being the biggest value. For example I got $P(U = 1) = 1/1296$ and $P(U = 6) = 671/1296$, which is obviously wrong.
 A: Some hints. 
If you roll a dice four times you obtain a four digit number, so there are $6^4$ different outcomes. How many outcomes will result in having $6$ as the minimum value? Clearly only one, namely $6666$. So $P(U=6)=\frac{1}{6^4}$.
How many outcomes are there where $5$ is the minimum value? Well one posibility is that $5$ is rolled on the first roll, i.e. the outcome is of the form $5xyz$, where $5\leq x,y,z \leq 6$. There are clearly $2^3$ outcomes of this form. Another possibility is that the minimum $5$ is rolled not on the first but on the second roll, i.e. the outcome is of the form $65xy$, where $5\leq x,y \leq 6$. There are clearly $2^2$ outcomes of this form. Similarly there are $2$ outcomes where the minimum $5$ is rolled first on the third roll and only one where it is rolled first on the last roll. So there are $2^3 + 2^2 + 2 + 1=15$ outcomes where $5$ is the minimum value. So $P(U=5)=\frac{15}{6^4}$.
You may continue like this. There are probably easier ways of doing this exercies. 
