बेज़ प्रमेय: परिभाषा और उदाहरण

Bayes' Theorem

• Describes the probability of an event based on prior knowledge of conditions related to the event.
• Expressed as: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
• $P(A|B)$: Posterior probability of A given B is true.
• $P(B|A)$: Probability of B given A is true.
• $P(A)$: Prior probability of A.
• $P(B)$: Prior probability of B.

Deduction

• Derived from conditional probability definitions: $P(A|B) = \frac{P(A \cap B)}{P(B)}$ and $P(B|A) = \frac{P(B \cap A)}{P(A)}$
• Since $P(A \cap B) = P(B \cap A)$: $P(A|B)P(B) = P(B|A)P(A)$
• Dividing by $P(B)$ yields Bayes' Theorem.

Example: Disease Detection Test

• Probablity of a person having the disease $P(A)$ = 0.01 or 1%
• Test correctly identifies disease presence $P(B|A)$ = 95% or 0.95
• Test incorrectly indicates disease absence $P(B|\neg A)$ = 5% or 0.05
• Goal: Probability of having the disease given positive test $P(A|B)$.
• Calculate $P(B)$: $P(B) = P(B|A)P(A) + P(B|\neg A)P(\neg A)$
• $P(\neg A) = 1 - P(A) = 0.99$
• $P(B) = (0.95 \times 0.01) + (0.05 \times 0.99) = 0.059$
• $P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{0.95 \times 0.01}{0.059} \approx 0.161$
• Even with a positive test, the likelihood of having the disease is only about 16.1%, emphasizing the impact of prior probabilities.

Thermodynamics

• Relates macroscopic properties of a system using laws and mathematical reasoning.
• Classical thermodynamics concerns macroscopic, not microscopic, properties.
• Volume, Pressure, Temperature, and Density are examples of macroscopic properties

Defintions

• System: A defined quantity of matter separated from its surroundings by a boundary.
• Property: A macroscopic system characteristic with an assignable numerical value, independent of past behavior.
• State: The condition of a system defined by its properties.
• Process: Change in a system's state.
• Equilibrium: State where system properties do not change with time.
• Property: A macroscopic characteristic of a system to which a numerical value can be assigned at a given time without knowledge of the system's previous behavior.
• State: The condition of a system as described by its properties.
• Process: A change in the state of a system.
• Equilibrium: A system is in equilibrium if its properties are not changing with time.

System Types

• Closed: Fixed mass.
• Open: Mass crosses the boundary.
• Isolated: No interaction with surroundings.

Equilibrium Types

• Thermal: Uniform temperature.
• Mechanical: Uniform pressure.
• Phase: Constant mass of each phase.
• Chemical: Constant chemical composition.

Units

SI Units

• Mass: kg (kilogram).
• Length: m (meter).
• Time: s (second).
• Temperature: K (Kelvin).

English Units

• Mass: lb (pound).
• Length: ft (foot).
• Time: s (second).
• Temperature: °R (degrees Rankine).

Conversion Factors

• 1 lb = 0.453592 kg
• 1 ft = 0.3048 m

Specific Volume and Density

• Specific Volume: $v = \frac{V}{m}$ (V = volume, m = mass)
• Density: $\rho = \frac{1}{v} = \frac{m}{V}$

Pressure

• Definition: $P = \frac{F}{A}$ (F = force, A = area)

Pressure Units

• Pascal: $\frac{N}{m^2}$
• 1 bar = $10^5 Pa$
• 1 atm = $1.01325 \times 10^5 Pa = 14.696 \frac{lbf}{in^2}$

Temperature conversion

• $T(^{\circ}C) = T(K) - 273.15$
• $T(^{\circ}F) = T(^{\circ}R) - 459.67$
• $T(^{\circ}R) = 1.8T(K)$
• $T(^{\circ}F) = 1.8T(^{\circ}C) + 32$

Example: Car Tire Pressure

• Gauge pressure = 32 lbf/in², atmospheric pressure = 14.7 lbf/in².
• Use: $P_{abs} = P_{gauge} + P_{atm}$
• Absolute pressure: $P_{abs} = 32 \frac{lbf}{in^2} + 14.7 \frac{lbf}{in^2} = 46.7 \frac{lbf}{in^2}$
• Convert to kPa: $P_{abs} = 46.7 \frac{lbf}{in^2} \times \frac{1.01325 \times 10^5 Pa}{14.696 \frac{lbf}{in^2}} \times \frac{1 kPa}{1000 Pa} = 322 kPa$

Complex Numbers

• Form: $a + bi$
• a and be are real numbers
• i is the unit i = $\sqrt{-1}$

Definitions

• a is the real part
• b is the imaginary part

Operations

Addition:

• $\qquad (a + bi) + (c + di) = (a + c) + (b + d)i$
• Example: $(2 + 3i) + (1 - i) = 3 +2i$

Subtraction

• $\qquad (a + bi) - (c + di) = (a - c) + (b - d)i$
• Example: $(2 + 3i) - (1 - i) = 1 +4i$

Multiplication:

• $\qquad (a + bi)(c + di) = (ac - bd) + (ad + bc)i$
• Since $i^2 = -1$
• Example: $(2 + 3i)(1 - i) = 5 + i$

Division:

• Multiply the numerator and denominator by the conjugate of the denominator.
• $\qquad \frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i$
• Example: $\frac{2 + 3i}{1 - i} = -\frac{1}{2} + \frac{5}{2}i$

Complex Conjugate

• Complex conjugate of $a + bi$ is $a - bi$, denoted as $\overline{a + bi}$.
• Property: $z + \overline{z} = 2Re(z)$, where $Re(z)$ is real part of $z$.
• Property: $z - \overline{z} = 2iIm(z)$, where Im(z) is imaginary part of z.
• Property: $z\overline{z}$ is a real number.

Modulus (Absolute Value)

• Modulus of z = a+bi:
• $\qquad |z| = \sqrt{a^2 + b^2}$
• Properties:
  • $|z| \geq 0$
  • $|z| = 0$ if and only if $z = 0$
  • $|z_1 z_2| = |z_1| |z_2|$
  • $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} , z_2 \neq 0$

Argument

• Argument of z = a+bi :
• $\qquad \theta = \arg(z) = \arctan(\frac{b}{a})$
• Note: Adjust the angle based on the quadrant of (a,b).
• Principal arument is in the value of $\theta$ in the interval $(-\pi, \pi]$

Polar Form

• Polar Form: z = r(\cos(\theta) + i\sin(\theta)).
• r = |z| (modulus).
• $\theta = \arg(z)$ (argument).

Euler's Formula

• Euler's Formula: $e^{i\theta} = cos(\theta) + i\sin(\theta)$.

Exponential Form

• Exponential From: z = re^{i\theta}

De Moivre's Theorem

• For any $z = r(\cos(\theta) + i\sin(\theta))$ and any integer $n$: $z^n = r^n (\cos(n\theta)+i\sin(n\theta))$ or in exponential form: $(re^{i\theta})^n = r^ne^{in\theta}$

DataRobot

Overview

• DataRobot: AI lifecycle platform (build, deploy, monitor, manage).

Benefits

• Faster Time to Value.
• Better Predictions.
• Actionable Insights.
• Lower Total Cost of Ownership.

Key Differentiators

• Automation and Transparency.
• Best-in-Class Accuracy.
• AlOps and Governance.
• Open and Flexible Platform.

DataRobot AI Platform Capabilities

• Automated ML: Rapid model building, optimizes accuracy.
• MLOps: Model management and monitoring, assures accuracy.
• Decision Intelligence: Improves decision with powered insights.
• Data Engineering: Transform raw data into high-quality features.

DataRobot Use Cases

Industry

• Financial Services, Healthcare, Retail, Manufacturing

Department

• Marketing, Sales, Supply Chain, Risk Management

Business Challenges

• Customer Churn, Fraud Detection, Predictive Maintenance, Demand Forecasting

DataRobot Architecture

Data sources -> Data Engineering -> Automated ML -> MLOps -> Decision Intelligence -> Business Applications -> API Integrations

Partial Differential Equations

Definition

• Equations involving multivariable functions and partial derivatives, used in physics and engineering.

Examples

• Wave equation: describes waves $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}.$
• Heat equation: describes heat transfer $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$.
• Laplace equation: describes equilibrium states $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

Classification of 2nd Order Linear PDEs

Describes general form and discriminant types of PDEs.

• General form: $ A\frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x\partial y} + C\frac{\partial^2 u}{\partial y^2} + D \frac{\partial u}{\partial x}+ E \frac{\partial u}{\partial y} + Fu = G.$
• Discriminant $\Delta = B^2 - 4AC $. - Hyperbolic: $\Delta > 0 $, - Parabolic: $\Delta = 0$, - Elliptic: $\Delta < 0.$

Boundary Conditions

• Dirichlet: Specifies solution value on boundary.
• Neumann: Specifies the solutions of normal derivative on boundary.
• Robin: Specifies linear combination of the solutions of normal derivative on boundary.

Solution Techniques

• Analytical methods, Separation of variables, Fourier series, Green's functions
• Numerical methods, Finite difference, Finite element, Finite volume

Applications

• Physics, Fluid dynamics, Electromagnetism, Quantum mechanics
• Engineering, Heat transfer, Structural analysis, Signal processing
• Finance, Option pricing, Risk management.

Software Packages

• Open source: FEniCS, deal.II, OpenFOAM
• Commercial: COMSOL, ANSYS, MATLAB

Linear Algebra (in French)

Systems of Linear Equations

• A system of m linear equations with n unknowns $ x_1, x_2, \dots, x_n :$ $$ \begin{cases} a_{11}x_1 + a_{12}x_2 +.... + a_{1n}x_n = b_1 \
  a_{21}x_1 + a_{22}x_2 +.... + a_{2n}x_n = b_2 \
  .... \
  a_{m1}x_1 + a_{m2}x_2 +.... + a_{mn}x_n = b_m \end{cases} $$
• Solving a system of linear equations: finding all the possible values of the unknowns
• Methods of resolving -Substitution -Elimination Gauss -Matrix Method

Matrices

• A matrix is a table of numbers (reals or complexes) organized in m rows n Columns.

Operations on matrixe

• Addition A + B ( if A and B have the same size) -Scalar Matrix: lambda A
• Multiplication of matrices: A*B -Transposition: A^T

Determinants

• Associates a scalar with a square matrix.
• Denoted as det(A) or |A|.
• Null if line is 0.
• Matrix multiplication = lambda

Vector Spaces (Espaces vectoriels)

• Set of objects (vectors) with addition and scalar multiplication operations.

Linear mappings (Applications linéaires)

• A Function f between 2 vector spaces
• Must perserve vector value.

Eigenvalues and Eigenvectors (Valeurs propres et vecteurs propres)

• A vector v such as: Av = lambda v. It is assoicated with scalar value.
• Diagonalization= finding invertible matrix P

Economics

Definition

• The study of how societies use limited resources to produce valuable goods and services and how they are distributed.

Microeconomics vs Macro economics

• Micro focuses on individuals, consumers, business
• Macro focuses on the larger picture

Core Principles

• Scarcity
• Limited resources against unlimited needs.
• Opportunity Cost
• Value forgone in the next best decision.
• Incentives
• People respond to incentives.

Economic Systems

• Market Economy
• Decisions made by individuals and firms.
• Planned Economy
• The govenment makes decisions.
• Mixed Economy
• Mix of Market and Planned

Demand And Supply

• Demand
• The customers purchase.
• Supply The producers offer.
• Market equilibrium equal

Elasticity

• Price Elasticity Of Demand
• How sensible a purchaser is to a change in price.

Market Dynamics

• Perfect competition, Monopoly, Oligopoly, Monopolistic Competition

Macroeconomics

• GDP
• Inflation
• Unemployment
• Fiscal Policy
• Monetary Policy

Bernoulli's Principle

• Discovered by Daniel Bernoulli during the 18th century. States that for an inviscid flow of a nonconducting fluid, an increase in fluid speeds occurs simultaneously.

Equations

$\frac{v^2}{2} + gz + \frac{p}{\rho} = \text{constant}$

Where:

• $v$ is the fluid flow speed at a point on a streamline
• $g$ is the acceleration due to gravity
• $z$ is the elevation of the point above a reference plane
• $p$ is the pressure at the point
• $\rho$ is the density of the fluid

Simplified

$p + \frac{1}{2} \rho v^2 = \text{constant}$

Venturi Meter

• Venturi Meter: A device that measures the flow rate of a fluid through a pipe.

Equation

$Q = A_1 \sqrt{\frac{2(P_1 - P_2)}{\rho(r^2 - 1)}}$

• $Q$: flow rate,
• $A_1$: cross at point 1.
• $A_2$: cross section in middle narrowed part.
• $P_1$: pressure at point 1.
• and middle pressure.
• $\rho$: fluid density.

Technical Indicators

• Technical Indicators are calculations based on historical data to find changes to price.
• Divided into two types.
• Overlay- Appears on the price of graph. Oscillator=A graph that is sperate from the price graph.

Types of use

• Determine Trends
• Generate Market Signals Confirm price patterns

SMA

• Simple moving average calculates the avergae of a specified time.

$\qquad SMA = \frac{\sum_{i=1}^{n} Precio_i}{n}$ Interpretation

• Price Over SMA equals trend.
• Prive under SMA equal down trend.

EMA

• Exponetial moving average- places weigth onto recent average

MACD

• Relation between 2 exponential values.

$\qquad MACD = EMA_{12} - EMA_{26}$

$\qquad Señal = EMA_9(MACD)$

$\qquad Histograma = MACD - Señal$

• Divergence, the price might start to drop soon.

Relative force indez

• Assess over purchased and over sold indicators.

Equation

$\qquad RSI = 100 - \frac{100}{1 + RS}$

Dónde:

$\qquad RS = \frac{Promedio\ de\ ganancias}{Promedio\ de\ pérdidas}$

Interpretation

• Over purchase >70 indicates a price correction.
• Over sold <30, May go up. Diverge= Indicates that a downwards trend is diminishing.

Fibonacci numbers

• Identify support/resistance levels.