Experiments, Sample Space and Events

• Experiment: A process producing an outcome.

  • Deterministic Experiment: Outcome known in advance (e.g., stretching a spring with a weight and measuring its length).
  • Random Experiment: Outcome not known in advance (e.g., tossing a coin and observing the face).

• Sample Space: The set of all possible outcomes ($\Omega$).

  • Sample Point: Each outcome $\omega \in \Omega$.
  • Discrete Sample Space: Finite or countable outcomes.
    • Example: Tossing a coin and observing the face. $\Omega = {H, T}$ or multiple tosses until a heads appears $\Omega = {H, TH, TTH, TTTH,...}$
  • Continuous Sample Space: Uncountably infinite outcomes.
    • Example: Height of a student in cm. $\Omega = [0, 300]$

• Event: Any subset of the sample space ($\Omega$).

  • Occurs if the outcome is an element of the event.
  • Elementary Event: Contains only one outcome. Example: ${H}$.
  • Certain Event: Equal to the sample space ($\Omega$). Example: ${H, T}$.
  • Impossible Event: Empty set ($\emptyset$). Example: Observing heads and tails at the same time.

Set Operations

• Complement: $A^c$ outcomes not in A. $A^c = {\omega \in \Omega: \omega \notin A}$

• Union: $A \cup B$, outcomes in A or B or both. $A \cup B = {\omega \in \Omega: \omega \in A \text{ or } \omega \in B}$

• Intersection: $A \cap B$, outcomes in both A and B. $A \cap B = {\omega \in \Omega: \omega \in A \text{ and } \omega \in B}$

• Disjoint/Mutually Exclusive: $A \cap B = \emptyset$

• Example: Tossing a die. $\Omega = {1, 2, 3, 4, 5, 6}$. $A = $ observing an even number ${2, 4, 6}$, $B =$ observing a number greater than $3$ ${4, 5, 6}$, $A^c = {1, 3, 5}$, $A \cup B = {2, 4, 5, 6}$, and $A \cap B = {4, 6}$.

Probability Definition

• Probability Function, $P$: assigns a real number $P(A)$ to each event $A$.

1. Non-negativity: $P(A) \geq 0$, for all events A.
2. Normalization: $P(\Omega) = 1$.
3. Additivity: For mutually exclusive events $A_1, A_2, A_3,...$, $P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i)$.

• $P(A)$ is the probability of the event.

Probability Properties

• $P(\emptyset) = 0$

• $P(A^c) = 1 - P(A)$

• If $A \subseteq B$, then $P(A) \leq P(B)$

• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• $P(A \cup B) \leq P(A) + P(B)$

• Example: Tossing a fair coin twice. $\Omega = {HH, HT, TH, TT}$. Each outcome with probability $\frac{1}{4}$. Let A be the event of observing at least one head, then $A = {HH, HT, TH}$, $P(A) = \frac{3}{4}$. Let B be the event of observing two heads, then $B = {HH}$, $P(B) = \frac{1}{4}$.

Computing Probabilities

• Discrete Sample Space: Probability of any event A is $P(A) = \sum_{\omega \in A} P(\omega)$.
• Equally Likely Outcomes: $P(A) = \frac{|A|}{|\Omega|}$.
• Example: Tossing a fair die. $\Omega = {1, 2, 3, 4, 5, 6}$. $A =$ observing an even number ${2, 4, 6}$. $P(A) = \frac{3}{6} = \frac{1}{2}$.

Algorithmic Complexity

• Measures time (time complexity) and space (space complexity) used by an algorithm for input of size $n$.
  • Time Complexity: function of the time amount taken as input grows longer.
  • Space Complexity: function of the memory space taken as input grows longer.
  • Expressed using Big O notation.

Big O Notation

• Describes the limiting behavior of a function as the argument tends towards a particular value or infinity.

Common complexities

Example: Find the maximum element in an array Time complexity is $O(n)$.

Algorithmic Game Theory

• Game theory: Strategic interactions among multiple agents in a system.
  • Each agent's outcome depends on others' actions.
  • Applications: Economics, politics, computer science.

Selfish Routing

- System: Graph $G = (V, E)$.
- $r$ agents/players.
- Each agent $i$ controls $w_i$ amount of flow.
- $s_i, t_i \in V$: Agent $i$ wants to route $w_i$ from $s_i$ to $t_i$.

• Definition 1: Flow vector $f$.
  • $f_e =$ amount of flow on edge $e$.
  • $f_e = \sum_{i: e \in P_i} w_i$
• Definition 2: Cost on edge $e$.
  • $l_e(f_e)$.
  • $l_e(\cdot)$ is a cost function.
  • Often assumes that it is non-decreasing.
• Cost of path $P_i$:
  • $c_{P_i}(f) = \sum_{e \in P_i} l_e(f_e)$
• Goal: Flow vector $f$ that minimizes $\sum_{i=1}^r c_{P_i}(f) = \sum_{e \in E} f_e \cdot l_e(f_e)$

Braess's Paradox

• Setting: 1 unit of flow from $s$ to $t$ with 2 paths.
• Demonstrates that adding a new road can worsen overall traffic flow.

Price of Anarchy

• Social optimum flow $f^*$: Minimizes total cost.
• User equilibrium flow $f$: No agent can improve its cost unilaterally.
• $PoA = \frac{\text{Total cost of user equilibrium flow } f}{\text{Total cost of social optimum flow } f^} = \frac{\sum_{e \in E} f_e \cdot l_e(f_e)}{\sum_{e \in E} f_e^ \cdot l_e(f_e^*)}$

Finite Element Method

• Numerical technique solving problems formed as functional minimization or PDEs.
• Represents domain as assembly of finite elements.
• Uses approximating functions to represent solution.

Typical Uses:

- Stress Analysis
- Heat Transfer
- Fluid Flow
- Electromagnetic Potential

Simple Example $\frac{d^{2}u}{dx^{2}}+u+x=0, \quad 0

What is %

• A way to express number as fraction of 100
• Symbol %
• Ex: 25% means 25/100

How to calculate percent of number?

• Multiply number by percent expressed as fraction/decimal

  • Example: Calculate 20% of 50
  1. Convert percent to fraction: 20% = 20/100
  2. Multiply number with fraction: 20/100 * 50 = 10

• 20% of 50 is 10

How to transform number in %

• Multiply number by 100. Ex: Transform 0.75 in %: 0.75 * 100 = 75% (0.75 is equal to 75%)

Physics: Vectors

Vector Sum

• Graphical Method: Place vectors one after another, respecting module, direction, and sense.

  • The resulting vector unites the origin with the last extreme.

• Analytical Method: Decompose vectors into Cartesian, sum axis components.

  • If $\vec{a} = (a_x, a_y)$ and $\vec{b} = (b_x, b_y)$, Result is $\vec{R} = (a_x + b_x, a_y + b_y)$

Scalar Product

• $\vec{a}.\vec{b} = |\vec{a}|.|\vec{b}|.Cos (\alpha)$
• $\vec{a}.\vec{b} = a_x.b_x + a_y.b_y$

Vector Product

• $\vec{a} \times \vec{b} = |\vec{a}|.|\vec{b}|.Sen (\alpha). \hat{n}$
• $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_x & a_y & 0 \ b_x & b_y & 0 \end{vmatrix} = (a_x.b_y - a_y.b_x). \hat{k}$

Kinematics

Uniform Rectilinear Movement

v = change in space / change in time

Uniformaly Varied Rectilinear Movement

Dynamics

Newton's Laws

Work

Power

Potential Energy

Momentum

Física

Vectores

Suma de vectores

Método gráfico

• Se colocan los vectores uno a continuación del otro, respetando módulo, dirección y sentido.
• El vector resultante es aquel que une el origen del primero con el extremo del último.

Suma analítica de vectores

• Se descompone cada vector en sus componentes cartesianas, luego se suman las componentes de cada eje.
• If $\vec{a} = (a_x, a_y)$ and $\vec{b} = (b_x, b_y)$: $\vec{R} = (a_x + b_x, a_y + b_y)$

Producto escalar

$\vec{a}.\vec{b}$ = $|\vec{a}|.|\vec{b}|.Cos (\alpha)$. $\vec{a}.\vec{b}$ = $ a_x.b_x + a_y.b_y$

Producto vectorial

$\vec{a} \times \vec{b} = |\vec{a}|.|\vec{b}|.Sen (\alpha). \hat{n}$ $\vec{a} \times \vec{b} = (a_x.b_y-a_y.b_x) \hat{k}$

• Python is a popular programming language.
• Created by Guido van Rossum, released in 1991.

Used for:

- web development (server-side),
- software development,
- mathematics,
- system scripting.

Can do:

• can be used in server to create web apps
• can modify files
• can be used to handle data
• can be scripted
• Python works on different platforms (Windows, Mac, Linux, Raspberry Pi, etc).
• Python has a simple syntax similar to the English language.
• Python has a syntax that allows developers to write programs with fewer lines than some other programming languages.
• Python is interpreted, meaning that code can execute as soon as it is written. This means that prototyping can be very fast.
• Python can be handled in a procedural way, oriented to objects, or as a functional.
• newest python is 3

ESE 204 Signals & Systems

Complex Power

• Electronics and signal processing often deal with AC circuits/signals.
• Complex Power: A time-independent quantity with information of the power in AC circuits.

Complex Power Definition

Let $v(t) = V_m cos(\omega t + \theta_v)$ and $i(t) = I_m cos(\omega t + \theta_i)$ $\tilde{V} = V_m e^{j\theta_v}$ $\tilde{I} = I_m e^{j\theta_i}$ $S = \frac{1}{2} \tilde{V} \tilde{I}^* = \frac{V_m I_m}{2} e^{j(\theta_v - \theta_i)}$, $\tilde{I}^*$ is the complex conjugate of $\tilde{I}$ $S = P + jQ$, Where P is the average power and Q is the reactive

• Units : Volt-Amperes. (VA) Average Power: Watts (W) and reactive power: Volt-Amperes reactive (VAR)

Power factor

$pf = cos(\theta_v - \theta_i)$. Number between -1 and 1

• Lagging: If Current lags the voltage (Inductive Load)
• Leading: If Current leads the voltage (Capacitive Load)
• The power factor indicates how effectively the power is being used. High = effectively