[商管] [數統] 幾題頗為艱難的題型

[alex2014 (綠影)]

1. Let X be a discrete random variable whose range is the

   nonnegative integers . Show that

        ∞
   EX = Σ  (1-Fx(k)) , where Fx(k) = P(X≦k)
        k=0

2. A random variable X is defined by Z = logX, where E(Z)=0

   Is E(X) greater than ,less than , or equal to 1 ?


3. Does a distribution exist for which Mx(t)=t/(1-t),|t|＜1,

   If yes ,find it ,if no ,prove it ?


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→ stevegood:第一題觀念上可以寫幾項來看出，嚴謹的數學式要小技巧
→ stevegood:第二題隨便舉個例子好像是大於...
推 hookylen:第三題是不是卡方一
→ dawn90196:第二題我以為是等於耶@@"
推 dawn90196:第二題是大於等於 E(X)=E(e^Z)>=e^E(Z)=e^0=1
→ dawn90196:根據Jensen's不等式