Limits of integration for random variable Suppose you have two random variables $X$ and $Y$. If $X \sim N(0,1)$,  $Y \sim N(0,1)$ and you want to find k s.t. $\mathbb P(X+Y >k)=0.01$, how would you do this? I am having a hard time finding the limits of integration. How would you generalize $\mathbb P(X+Y+Z+\cdots > k) =0.01$? I always get confused when problems involve multiple integrals.  

Hint: Are the random variables independent?  
If so, you can avoid integration by using the facts 


*

*the sum of independent normally distributed random variables has a normal distribution

*the mean of the sum of random variables is equal to the sum of the means  

*the variance of the sum of independent random variables is the sum of the variances

*for a standard normal distribution $N(0,1)$: $\Phi^{-1}(0.99)\approx  2.326$  
