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Q10Section A (Section A)10 of 60

Maximum likelihood estimates in simple regression

Consider the linear regression model yi=β0+β1xi+ϵiy_i=\beta_0+\beta_1x_i+\epsilon_i; i=1,2,…,ni=1,2,…,n, where ϵi\epsilon_i's are i.i.d. standard normal random variables. Given that

1n∑i=1nxi=3.2,1n∑i=1nyi=4.2,1n∑j=1n(xj−1n∑i=1nxi)2=1.5 and\frac1n\sum_{i=1}^nx_i=3.2,\quad\frac1n\sum_{i=1}^ny_i=4.2,\quad\frac1n\sum_{j=1}^n\left(x_j-\frac1n\sum_{i=1}^nx_i\right)^2=1.5 \text{ and}

1n∑j=1n(xj−1n∑i=1nxi)(yj−1n∑i=1nyi)=1.7,\frac1n\sum_{j=1}^n\left(x_j-\frac1n\sum_{i=1}^nx_i\right)\left(y_j-\frac1n\sum_{i=1}^ny_i\right)=1.7,

the maximum likelihood estimates of β0\beta_0 and β1\beta_1, respectively, are

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