I have a problems solving the following:
Let $X_{1},...,X_{1500}$ identical independent distributed random variable with density function: $f(t)=\frac{3}{2}t^{2} \textbf{1}_{[-1,1]}(t)$
(1) Show: $\mathbb{E}(X_{1})=0$ and $Var(X_{1})=\frac{3}{5}$
(2) Calculate $\left(\sum_{i=1}^{1500} X_{i} \leq -33 \textbf{ or }\sum_{i=1}^{1500} X_{i} > 33\right)$ with usage of normal approximation without continuity correction.
So far i could calculate (1) (still i am not sure if it is correct):
$E(X_{1})=\int_{-1}^{1} x_{1} \frac{3}{2} x_{1}^{2} \mathrm{d}x = \frac{3}{2}\int_{-1}^{1} x_{1}^{3} = \frac{3}{2} \left[\frac{1}{4}x^{4}\right]_{1}^{-1}=\frac{3}{2}\cdot0=0$
For the variance i used: $Var(X_{1})=E(X_{1}^{2})-E(X_{1})^{2}$ and got as result $Var(X_{1})=\frac{3}{5}$ like in the task.
What i dont understand is how to calculate the probability with the normal approximation. Usually i have tasks like e.g. $P(10 < X < 12)$ but in this case i dont understand the or in the text. Can you help me solving this? For now i would be happy for any hint :-)