This document contains random formulae images I used in the notes.
\[ A = \{1, 2\} \] \[ B = \{1, 2, 3\} \]
\[ \begin{eqnarray} E[X^2] & = & \int_0^1 x^2 dx \\ & = & \left. \frac{x^3}{3} \right|_0^1 = \frac{1}{3} \end{eqnarray} \]
\[ \frac{|x - \mu|}{k\sigma} > 1 \] Over the set \( \{x : |x - \mu | > k\sigma\} \) \[ \frac{(x - \mu)^2}{k^2\sigma^2} > 1 \] \[ \frac{1}{k^2\sigma^2} \int_{-\infty}^\infty (x - \mu)^2 f(x) dx \] \[ \frac{1}{k^2\sigma^2} E[(X - \mu)^2] = \frac{1}{k^2\sigma^2} Var(X) \]
\[ P(A_1 \cup A_2 \cup A_3) = P\{A_1 \cup (A_2 \cup A_3)\} = P(A_1) + P(A_2 \cup A_3) \] \[ P(A_1) + P(A_2 \cup A_3) = P(A_1) + P(A_2) + P(A_3) \]
\[ P(\cup_{i=1}^n E_i) = P\left\{E_n \cup \left(\cup_{i=1}^{n-1} E_i \right) \right\} \]
\[ (x_1, x_2, x_3, x_4) = (1, 0, 1, 1) \] \[ p^{(1 + 0 + 1 + 1)}(1 - p)^{\{4 - (1 + 0 + 1 + 1)\}} = p^3 (1 - p)^1 \] \[ \mathrm{SD}(X) \mathrm{SD}(Y) \] \[ Var(X) \] \[ Var(X) = E[X^2] - E[X]^2 \rightarrow E[X^2] = Var(X) + E[X]^2 = \sigma^2 + \mu^2 \] \[ Var(\bar X) = E[\bar X^2] - E[\bar X]^2 \rightarrow E[\bar X^2] = Var(\bar X) + E[\bar X]^2 = \sigma^2/n + \mu^2 \] \[ f(x | y = 5) = \frac{f_{xy}(x, 5)}{f_y(5)} \]
\[ P(A\cap B) \] \[ P(A) \] \[ P(A\cap B^c) \]
\[ \frac{10!}{1!9!} = \frac{10\times 9 \times 8 \times \ldots \times 1}{9 \times 8 \times \ldots \times 1} = 10 \]
\[ \frac{10!}{2!8!} = \frac{10\times 9 \times 8 \times \ldots \times 1}{2 \times 1 \times 8 \times 7 \times \ldots \times 1} = 45 \]
In general
\( \left(\begin{array}{c}n \\ 2\end{array}\right)= \frac{n \times (n - 1)}{2} \)
\[ \mu \]
\[ \sigma^2 \]
\[ E[Z] = E\left[\frac{X - \mu}{\sigma} \right] = \frac{E[X] - \mu}{\sigma} = 0 \]
\[ Var(Z) = Var\left(\frac{X - \mu}{\sigma}\right) = \frac{1}{\sigma^2} Var(X - \mu) = \frac{1}{\sigma^2} Var(X) = 1 \]
\[ E[X_i^2] = E[Y_i] = \sigma^2 + \mu^2 \] \[ \sum_{i=1}^n (X_i - \bar X)^2 = \sum_{i=1}^2 X_i^2 - n \bar X ^ 2 \]
\[ E[\chi^2_{df}] = df \] \[ E[S^2] = \sigma^2 \rightarrow E\left[\frac{(n-1)S^2}{\sigma^2}\right] = (n-1) \] \[ Var(\chi^2_{df}) = 2df \]