Question

6. This problem considers the simple linear regression model, that is, a model with a single covariate r that has a linear relationship with a response y. This simple linear regression model is y = Bo + Bix +, where Bo and Bi are unknown constants, and a random error has normal distribution with mean 0 and unknown variance o' The covariate a is often controlled by data analyst and measured with negligible error, while y is a random variable. (a) What is the pdf of y? (b) Find E(y) and Var(y). Since the coefficients Bo and B, are unknown, we estimate them by using n pairs of data. sa (11.21), .... ( n). Thus, we rewrite the simple linear regression Yi = Bo + B13; + i = 1,...,n, Where is identically independently distributed from normal distribution with men and unknown variance o'. The least squares estimators of Bo and Bisay Bo and Bi respectively are defined as Bo = ý ßit, where 3 = 4 3 7 = m, and S = (-13 - )". (c) Show that ) = 0 and = 1. (d) Show that E(61) = B, and E(X) = Bo. (e) Show that Var(i) = (1) Show that Covly. 6) = 0.

Answer 1

To simple linear regression model y = ß + B xt E lathine , & NN (0,02) BB are unknown 2 . i8 non stochastic dhe x is also constant -Flamown quoustily since y depends upon { which if normally distributed so y is also follow a normal distribution with Bly) - El B+BX+ Ed] I wing n pair's of data 4* =ß+BX+ Eis un Given that lecust-square estimate of R = 5-Å 6 ( ). where sa Ŝ ( 36 -2012 6 - +x+o pelkoa), fox-7) [Ely) = Pt B x ans. and variyda ver (37827) | -Lex - 2) T = s[mx-ne] varey) = 0 Jan - hy NM (772, 02) Fuxia) @ff of y 3 : - Varca) s =0 |f04) te exp-(92478217 21 r Ans.

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