$X \sim$ Uniform$(0, 1)$ and $Y\sim$ Uniform$(0, 1)$ compute $P(X+Y\ge0.5)$
Disclaimer: In this homework-type question, $X$ and $Y$ are probably assumed to be statistically independent. My answer assumes this, although you didn't state it explicitly.
State the joint PDF of $X$ and $Y$ (hint: it's constant $1$ in a square and $0$ otherwise) and calculate $1 - P(X + Y < 0.5)$ instead. Identify the area of duples in the square $(x,y) \in [0,1]^2$ that fulfill $x + y < 0.5$ and calculate the probability by calculating the area of the resulting trianlge (easy).
P.S.: As mentioned by Henry, your assumption $N(0,2)$ is bogus indeed.