Constructing PDA to accept language $\{a^ib^j \mid 0 \leq j \leq 2i\}$
The description of the PDA is easy:


  Read the input string. if read $b$ and the stack is empty, reject the string. If see $a$, push two $a$ to the stack. If read $b$, pop a value from the stack. Finally, if read the end of the string, it will be accepted.