Formulas

These are all of the important formulas for this course. All of these will be on the back page of both the midterm and final exams.

Probability

Let \(X\) be a discrete random variable, \(x_i\) a potential outcome for \(X\), and \(p_i\) the probability that outcome occurs. Then:

Expected value of a discrete random variable: \(E[X] = \mu_X = \sum_{i=1}^n x_i p_i\)
Variance of a discrete random variable: \(Var(X) = \sigma^2_X = E[(X - \mu_X)^2] = \sum_i (x_i - \mu_X)^2 p_i\)
If \(Y\) is another random variable, \(Cov(X, Y) = \sigma_{XY} = E[(X - \mu_X)(Y - \mu_Y)]\)
Correlation of two random variables: \(\rho_{XY} = \frac{\sigma_{XY}}{\sqrt{\sigma_X^2\sigma_Y^2}}\)

Let \(X\) be a random variable and let \(x_i\) be an observation of a sample of \(X\).

The estimator of the expected value of \(X\) is the sample mean: \(\bar{x} = \frac{1}{n} \sum_{i = 1}^n x_i\)
The estimator for \(Var(X)\) is \(\frac{1}{n-1}\sum_{i=1}^n(x_i - \bar{x})^2\)
The estimator for \(Cov(X, Y)\) is \(\frac{1}{n-1}\sum_{i=1}^n(x_i - \bar{x})(y_i - \bar{y})\)

The true model: \(y_i = \beta_0 + \beta_1 x_i + u_i\)

The estimated model: \(y_i = \hat{\beta_0} + \hat{\beta_1}x_i + e_i\)

Formulas for simple regression coefficients:

\[\hat{\beta_0} = \bar{y} - \hat{\beta}_1 \bar{x}\]

\[\hat{\beta_1} = \frac{(\sum_i x_i y_i) - \bar{x}\bar{y}n}{(\sum_i x_i^2) - \bar{x}^2 n}\]

The \(R^2\) of a regression: \(\frac{\sum_i (\hat{y}_i - \bar{y})^2}{\sum_i (y_i - \bar{y})^2} = 1 - \frac{\sum_i e_i^2}{\sum_i (y_i - \bar{y})^2}\)