Let 𝑋 and 𝑌 be independently random variables, with 𝑋 uniformly distributed on [0,1] and 𝑌 uniformly distributed on [0,2] . Find the PDF 𝑓𝑍(𝑧) of 𝑍=max{𝑋,𝑌} .

Question

For 𝑧<0 or 𝑧>2 :

𝑓𝑍(𝑧)=

For 0≤𝑧≤1 :

𝑓𝑍(𝑧)=

For 1≤𝑧≤2 :

𝑓𝑍(𝑧)=

I think for the first scenario is the PDF is 0 since the bounds are outside of the range.

if anybody got hint for the other two, thanks