Let X be a positive random variable with distribution function F. Show that E(X)=∫∞0(1−F(x))dx
Attempt
∫∞0(1−F(x))dx=∫∞0(1−F(x)).1dx=x(1−F(x))|∞0+∫∞0(dF(x)).x (integration by parts)
=0+E(X) where boundary term at ∞ is zero since F(x)→1 as x→∞
Is my proof correct?
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