Statistics Solutions

[Solution] Let Y_1, ..., Y_n be i.i.d. random variables with pdf given by the Standard Normal distribution. Suppose this random sample is randomly split

Question: Let \(Y_{1}, \ldots, Y_{n}\) be i.i.d. random variables with pdf given by the Standard Normal distribution. Suppose this random sample is randomly split into two parts, one part with \(k\) observations and the second part with \(n-k\) observations. Define the sample means of these two parts respectively as

\[\bar{Y}_{k}=\frac{1}{k} \sum_{i=1}^{k} Y_{i}\]

and

\[\bar{Y}_{n-k}=\frac{1}{n-k} \sum_{i=k+1}^{n} Y_{i}\]
  1. Find the distribution of \(\frac{1}{2}\left(\bar{Y}_{k}+\bar{Y}_{n-k}\right)\).
  2. Find the distribution of \(k\left(\bar{Y}_{k}\right)^{2}+(n-k)\left(\bar{Y}_{n-k}\right)^{2}\)

(Hint: Where possible, use standard results to answer this question.)

Price: $2.99
Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document


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