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.
Limits of integration for random variable
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statistics
1 Answers
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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 $N(0,1)$: $\Phi^{-1}(0.99)\approx 2.326$
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1And if they are _not_ independent, then knowing the distribution of each of them is not enough anyway. – 2011-11-13