Problems

Problem 1. Calculate $3 + 4^2$.

Solution. $4^2 = 4\times 4 = 16$, therefore the answer is $3 + 16 = 19$.

Problem 2. If $\mathfrak A = 3$ and $\mathbb{B} = -2$, find $\mathfrak A + \mathbb{B}$.

Solution. $3 + (-2) = 1$.

Goals

Students should know:

How to count numbers
How to add and subtract
How to multiply and divide

\[\label{eq:special} \mathrm{d}W^2(t) = 2W(t)\mathrm{d}W(t) + \mathrm{d}t.\] This is known from [1].

These are our Examples

Math formulas

The mapping $\mathcal F : C(\mathbb R) \to \mathbb R$ is defined as $\mathcal F : F \mapsto F(0)$. The image $\mathfrak F := \mathcal F(C(\mathbb R))$ coincides with $\mathbb R$. We can use different notation in LaTeX such as $f, \mathbb f, \mathfrak f$, and all these are different letters.

\[\label{eq:new} x + y = z\] Consider the equation $\eqref{eq:new}$

Some Special Examples of Macros

$\RR$ is the set of all real numbers. Also it is denoted by $\BR$. $\bold{\alpha}$ is another example.

Picture!!

Brownian motion with drift g = 0.4 and diffusion \sigma = .5, starting from 0,
and the drift line x = gt — Brownian motion with drift $g = 0.4$ and diffusion $\sigma = .5$, starting from $0$, and the drift line $x = gt$

$M/M/1$ Queue in an Interactive Jump Environment

Consider an $M/M/1$ queue with an infinite waiting space operating in an interactive jump environment described as follows. Let $\mathcal{Z}$ be a finite or countable state space. We define a two-component Markov process $\mathcal{Y} = (X,Z)$ taking values in the countable state space $\mathbb Z_+ \times \mathcal{Z}$ with the following generator matrix $\mathbf{R}= \big( R[(n,z),(n',z')] \big)$: \[\begin{aligned} \label{Rmatrix-MM1} R[(n,z),(n+1, z)] &=& \alpha_z \lambda_z, \\ R[(n,z),(n-1, z)] &=& \alpha_z \mu_z, \\ R[(n,z),(n, z')] &=& \sigma_z \rho_z^{-n} \tau_n(z,z'), \\ R[(n,z),(n', z')] &=& 0, \non \\ R[(n,z),(n, z)] &=& - \sum_{(n',z') \in \mathbb Z_+ \times \mathcal{Z}\setminus \{(n,z)\}} R[(n,z),(n', z')], \non\end{aligned}\] where $\rho_z:= \lambda_z/\mu_z$ for each $z \in \mathcal{Z}$.

Consider the equation $\eqref{eq:special}$, which is known from .

Problem 3.3. \[\mathbb E X^2 = 1^2\cdot \frac16 + 2^2 \cdot \frac16 + \ldots + 6^2\cdot\frac16 = \frac{91}{6}\]

Problem 3.4. \[\Var X = \mathbb E(X^2) - (\mathbb E X)^2 = \frac{91}6 - \left(\frac72\right)^2 = \frac{35}{12}.\]

Problem 3.8. $\mathbb E (2X - 4) = 2\mathbb E X - 4 = 2\cdot\frac72 - 4 = -3$.

Problem 3.11. $\mathbb E(2X - 3Y) = 2\mathbb E X - 3\mathbb E Y = 2\frac72 - 3\frac72 = -\frac72$.