Suppose that $W_1$ and $W_2$ are i.i.d. and $P(W_i>x)=x^{-1/2}$ and $x$ is greater than or equal to $1$ and $i=1,2.$ How do you show that $P(W_1+W_2>x)=(2\sqrt{x-1})/x$? I know it involves calculus either differentiation or integration but I am not sure how to go about it. Any ideas?

Also, how do I use this above information for W to show that the value at risk (VAR alpha) is super-additive for all alpha values within 0 and 1 meaning that How do I show that VAR alpha(W1+W2)>VAR alpha(W1)+VAR alpha(W2). Any ideas?


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