Question

Problem 2: Let X be a random variable whose image X(S) is contained in the set {1, 2, ..., n}. Show that E(X) = () =Ë P(X k). k=1

Answer 1

Answer:-

Given That:-

Let X be a random variable whose image X(S) is contained in the set {1, 2, 3, ..., n}. Show that

E(X)= p(x2 k) k=1

Given,

X be a random variable whose image

X(S) is contained in the set {1, 2, 3, ..., n}

Such that

E(X)= p(x2 k) k=1

As we know that

$E(X)=\sum_{x=1}^{n}xPr(X=x)$

$E(X)=\sum_{x=1}^{n} \sum_{k=1}^{x} Pr(X=x)$

$E(X)=\sum_{k=1}^{n} \sum_{x=k}^{n} Pr(X=x)$ [By manipulating summation]

E(X)= p(x2 k) k=1

(this is also known tail sum formula)

Hence Proved

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