Let Xi be as in but with E(Xi ) = i and n1n i=1 i . Show

Chapter , Problem 2

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QUESTION:

Let \(X_i\) be as in Problem 1 but with \(E\left(X_{i}\right)=\mu_{i}\) and \(n^{-1} \sum_{i=1}^{n} \mu_{i} \rightarrow \mu\). Show that \(\bar{X} \rightarrow \mu\) in probability.

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QUESTION:

Let \(X_i\) be as in Problem 1 but with \(E\left(X_{i}\right)=\mu_{i}\) and \(n^{-1} \sum_{i=1}^{n} \mu_{i} \rightarrow \mu\). Show that \(\bar{X} \rightarrow \mu\) in probability.

ANSWER:

Step 1 of 4

We are given that  are independent random variables such that

 and . Also, we are given that

We need to prove that

(this means that  converges to  in probability).

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