bernoulli variable

• Binary variable
• \(X(v) = \begin{cases} 1 \text{ if } v=a_1 \\ 0 \text{ if } v = a_2\end{cases}\)
• PMF = \(P(X = x) = f(x) = \begin{cases}\rho_1 \text{ if } x = 1 \\ \rho_0 \text{ if } x = 0\end{cases}\)

Mean and Variance

Sample mean and variance

Binomial Distribution: Number of Occurrences

Multivariate Bernoulli Variable

Mean

Sample Mean

Covariance Matrix

Bivariate Analysis

mean

Sample mean

Sample Covariance Matrix

Attribute Dependence: Contingency Analysis

Contingency Table

• \(m_1 \times m_2\) matrix of observed counts \(n_{ij}\) for all pairs of values
• \(n_{ij}\) is the sum of objects that fit both i and j categories (shortjk)

\(X^2\) Statistic and Hypothesis Testing

• expected frequency \(= e_{ij} = n \cdot \hat\rho_{ij} = n \cdot \hat\rho^1_i \cdot \hat\rho^2_j = n \cdot \frac{n^1_in^2_j}{n}\)
• \(X^2 = \displaystyle\sum\limits^{m_1}_{i=1}\displaystyle\sum\limits^{m_2}_{j=1}\frac{(n_{ij}-e_{ij})^2}{e_{ij}}\)
• degrees of freedom = dof = q = S-t+1 = \((m_1-1)(m_2-1)\)
• m = # of columns or # rows respectively
• S = number of cells in contingency table (excluding margins)
• t = cells in margins
• on parameter is removed twice, so add 1 degree of freedom
• \(f(x|q) = \frac{1}{2^{q/2}\Gamma(q/2)} x^{\frac{q}{2}-1}e^{-\frac{x}{2}\)
• \(\Gamma(k>0) = \displaystyle\int\limits^\infty_0x^{k-1}e^{-x}dx\)

p-value

• if p-value < \(\alpha\) then reject null hypothesis
• if \(X^2 > z\) where \(pvalue(z) = \alpha\) then reject null hypothesis \(H_0\)

Multivariate Analysis

Covariance matrix

Multiway Contingency Analysis

\(X^2\) test