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I'm currently reading http://www.springer.com/mathematics/quantitative+finance/book/978-1-4614-4102-1 in which I stumbled upon the following:

For any nonnegative independent and identically distributed random variables $X, X_1,..., X_n$ it holds that

$$ P(X_1 +... X_n > x) = nP(X>x)P(X \leq x)^{n-1} + P(X_k > x \, \mathrm{and} \, X_l > x \, \mathrm{for \, some \, } k \neq l) + P(X_1 + ... + X_n > x \, \mathrm{and} \, X_k \leq x \, \mathrm{for \, every} \, k). $$

I get the first line since any one $X_k$ can be larger than $x$ while the rest are smaller. I also get the second line because any two $X_k$'s can be larger than $x$ while the rest are smaller. I even get the third line because the sum can be larger than $x$ even though all $X_k$'s are smaller than $x$.

What I don't get is why there is an equality sign. Shouldn't there be more cases covered by the right hand side, e.g., three $X_k$'s being larger than $x$ or ten $X_k$'s being larger than $x$ while the rest are smaller?

Any enlightenment is appreciated.

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1 Answer 1

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The equality signum is correct here because the term $\mathbb P\{X_k\gt x\mbox{ and } X_l\gt x\mbox{ for some }k\neq l\}$ means that there are at least two indexes $i_1$ and $i_2$ for which the $X_{i_1}$ and $X_{i_2}$ are greater than $x$. For example $\{X_1\gt x\}\cap\{X_2\gt x\}\cap \{X_3\gt x\}\subset \{X_k\gt x\mbox{ and } X_l\gt x\mbox{ for some }k\neq l\}$.

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I see. That explains it. Thank you very much! –  Fredric Sep 26 '13 at 13:56

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