Pstat 127 HW2.pdf - PSTAT 127 Winter 2018 Homework 2 Due in lecture Wednesday January 31 No late homework Throughout this homework clear working must be

# Pstat 127 HW2.pdf - PSTAT 127 Winter 2018 Homework 2 Due in...

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PSTAT 127, Winter 2018, Homework 2 - Due in lecture Wednesday January 31. No late homework. Throughout this homework, clear working must be shown to receive credit. The linear combination of independent normals theorem may help you in at least one part of this homework - see the hint file posted below homework 1 pdf on Gauchospace. (Note: if you didn’t see this theorem in the quarter you took PSTAT 120B, it may be proved using mgf’s. You are welcome to use the result without proof in this homework provided you understand the statement of this - I also am happy to work through the proof with anyone who wishes to see this.) 1. Suppose thatx1, . . . , xnare fixed constants, that1, . . . ,niidN(0, σ2)whereσ2is unknown, and thatY1, . . . , Ynarerandom variables satisfyingYi=βxi+i,i= 1, . . . , n.Note that this model does not include an intercept, and there is only one explanatory variable (i.e.,p= 1in the notationused in the class notes).(a) Write this model as a linear model in matrix form, including clear specifications of the dimensions and contents ofeach of the vectors/matricesY,X,β,in terms of the notation used above.Note thatβis a 1×1 matrix in this case (sincep= 1).Looking ahead to Monday Jan 29 lecture: the assumptions on vectorcan be written in terms of a multivariatenormal distribution asNn(0, σ2In). This notation will make sense after we cover the multivariate normaldistribution in lecture on Monday Jan 29 (not required in the rest of this homework).(b) Using the matrix formula derived in lecture for the ordinary least squares estimator ofβ(i.e., forˆβ)in the full