[問題] 一題poisson分配請教

[brucejune (...)]

Let X1,X2,X3 be iid Poisson random variable with parameter λ.

Let X4 be a Poisson random variable with parameter 2λ.

Which of the following statements MUST be true?

(a)The statement Cov(X1,X4)=2λ may be true.
(b)E(X1+X4)=3λ
(c)P(X1<X2<X3)=1/6
(d)X2+X4 is a Poisson random variable.

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想請教這題 除了(c)選項是完全不會之外 (b)選項確定對之外


(a)(d)選項只要X1,X2,X3與X4獨立就可以知道答案

但是要怎麼確認X1,X2,X3與X4獨立呢?(一直卡在這裡)


(c)選項 如果題目是continuous r.v.s 答案是1/6

但Poisson是discrete 所以這題該如何做呢?


麻煩版上高手幫忙解救!!

在此先謝謝了^^"

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→ Jordan23:X1,X2,X3與X4未必獨立, Cov(X,Y)≦sqrt(Var(X)Var(Y)).
→ brucejune:那可以請問(d)選項 跟(c)選項嗎  感激不盡!!
→ Jordan23:當然都是錯的!
→ brucejune:感謝j大!!
→ cation:P(Xi=Xj)>0,for all i,j=1,2,3 =>P(X1<X2<X3)<1/6