Distribution of function of two random variables Let $X$ be the number on a die roll, between 1 and 6. Let $Y$ be a random number which is uniformly distributed on $[0,1]$, independent of $X$. Let $Z = 10X + 10Y$.
What is the distribution of $Z$?

Hint:
$$
F_Z(x) = P(Z < x) = P(X + Y < x/10)
$$
Work it out by cases from here based on the potential values of $x$. For instance, if $x < 10$ then $x/10 < 1$, so $F_Z(x) = 0$ as $X \geq 1$. Another sample case: if $10 \leq x < 20$, then the value of $X$ in the right hand expression must be $1$ (in order for $X + Y < x/10$ to hold), and if $20 \leq x < 30$ then the value of $X$ must be either $1$ or $2$, each of which it takes with equal probability.
