# How do I prove that $E[X] = \int _{0}^\infty (1-F_x(t))dt$ [duplicate]

Area above CDF and below line is 1

• It's not quite clear what you intend by "Area above CDF and below line is 1", but this question has been answered mulitple times on site already. There's a one line proof in this answer and essentially the same one line proof (using different notation) in this answer – Glen_b -Reinstate Monica Nov 16 '17 at 3:13
• I'm new here is there any place that has tips for searching topics easily? – bleepblop Nov 16 '17 at 3:47
• Things might seem easy to you but for us beginners we take time... It's important that you understand that :( – bleepblop Nov 16 '17 at 3:52
• I get that it's quite possible you don't yet know how to search; "this has been answered" is still useful information. I searched integral survival function. If I hadn't found anything with that I had a few other search terms in mind. If I don't get anywhere with the built in search I do a site search via google (its search syntax is a little more powerful in some ways, and it also picks up information in comments, which is sometimes handy. The internal search has some nice features as well so between them they usually find something). – Glen_b -Reinstate Monica Nov 16 '17 at 6:24

suppose $X>0$
$$E(X)=\int _{0}^\infty xf(x)dx=-\int_{0}^\infty xd(1-F(x))=-x(1-F(x))|_{0}^\infty+\int _{0}^\infty [1-F(x)]dx=0+\int _{0}^\infty [1-F(x)]dx=\int _{0}^\infty [1-F(x)]dx$$