15.5.1 Two-dimensional random variables and their characteristics

Problem: The densities of uniformly distributed independent random variables \( X \) and \( Y \) are given: \( f_{1}(x)=1 \) in the interval \( (0 ; 1) \), outside this interval \( f_{1}(x)=0, f_{2}(y)=1 \) in the interval \( (0 ; 1) \), outside this interval \( f_{2}(y)=0 \). Find the distribution function and the distribution density of the random variable \( Z=X++Y \). Plot the graph of the distribution density \( g(z) \).