Tail-sum formula for continuous random variable

Let \(X\) be a positive random variable with c.d.f \(F\).

  1. Show using the representation \(X=F^{-1}(U)\) where \(U\) is \(\textrm{unif}(1,0)\) that \(\mathbf{E}(X)\) can be interpreted as the area above the graph on  \(y=F(x)\) but below the line \(y=1\). Using this deduce that
    \[\mathbf{E}(X)=\int_0^\infty [1-F(x)] dx = \int_0^\infty \mathbf{P}(X> x) dx \ .\]
  2. Deduce that if \(X\) has possible values \(0,1,2,\dots\) , then
    \[\mathbf{E}(X)=\sum_{k=1}^\infty \mathbf{P}(X\geq  k)\]

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