# Independent and Identically distributed random variables with value at risk [duplicate]

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?

• Add the self study tag. – Michael R. Chernick Feb 10 '19 at 3:15
• It seems you haven't started by searching (say on sum of random variables); there are numerous posts that discuss the procedure with independent variates (generally by convolution -- e.g. see this pdf). For example, see the following posts on our site ... – Glen_b -Reinstate Monica Feb 10 '19 at 6:03
• – Glen_b -Reinstate Monica Feb 10 '19 at 6:06
• If after reading the links (the last link in my first comment has numerous examples) you have a specific question, please post a new question. – Glen_b -Reinstate Monica Feb 10 '19 at 6:15
• Also note that you can use the fact that $F_{W_{i}}(w_i)=1-P(W_i\le w_i)$ and from here, pdfs can be worked out as well as joint pdfs given independence. – StatsStudent Feb 10 '19 at 6:16