Let Y₁, ..., Yₙ be independent and identically distributed (i.e., i.i.d.) random vari- ables with pdf given by the Standard Normal distribution N(0,1). 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
Yₖ= 1/kΣᵏᵢ₌₁Yᵢ and n Yₙ-ₖ= 1/ n-k Σⁿᵢ₌ₖ₊₁Yᵢ; Use arguments based on standard results (from Theorems in WMS, covered in the module notes, or designated Readings) to answer both parts of this question. 4 marks Find the distribution of 1/2 ( Yₖ + Yₙ-ₖ