如果X是非负整数值的,则EX=∑P{X>n}.谁能帮我证明一下

Question

如果X是非负整数值的,则EX=∑P{X>n}.谁能帮我证明一下

Answer 1

n>0时,有 P{x=n} = P{X>n-1}-P{x>n}
注意到 0*P{x=0} = 0,以下求和未注明的都从1到+∞求和
E(X) = ∑(n*P{x=n})
= ∑(n*(P{X>n-1}-P{x>n}))
= ∑(n*P{X>n-1}) - ∑(n*P{x>n})
= P{X>0} + ∑((n+1)*P{X>n}) - ∑(n*P{x>n})
= P{x>0} + ∑P{x>n}
= ∑P{x>n} （从0到+∞求和）