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=Chapter 17: Basic Statistical Models=

==Linear Regression==

For a bivariate data set <math>(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)</math>:

Assume that <math>x_1, x_2, \dots x_n</math> are not random

<math>y_1, y_2, \dots, y_n</math> are realizations of random variables <math>Y_1, Y_2, \dots, Y_n</math> that satisfy

<math>
Y_i = \alpha + \beta x_i + U_i
</math>

for <math>i=1 \dots n</math>

where <math>U_i</math> are independent random variables with <math>E(U_i) = 0</math> (because random fluctuations, expected to be zero about the regression line) and <math>Var(U_i) = \sigma^2</math> (each point has same variance, because assuming each random fluctuation has same amount of variability)

Expectation of each <math>Y_i</math> is different:

<math>
E[Y_i] = E[\alpha + \beta x_i + U_i] = \alpha + \beta x_i + E[U_i] = \alpha + \beta x_i
</math>

==Multiple Linear Regression==

If we considered that the data were better matched by a function like

<math>
y = \alpha + \beta x + \gamma x^2
</math>

then it's no longer ''linear regression'', it's ''multiple linear regression''

=Chapter 20=

==Mean Squared Error (MSE)==

Discussing unbiased estimators

Comparison of two unbiased estimators:

1. Variance (spread): less spread means better estimator

2. The lower the spread, the lower the MSE, the better the estimator







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