Scientific Notation and SI Units

Scientific Notation

• Used to express very large or very small numbers conveniently
• Expressed as $a \times 10^b$, where $1 \leq |a| < 10$ is a real number and $b$ is an integer
• Example: The speed of light is $3 \times 10^8 m/s$

International System of Units (SI)

• Standard system of units used in science and most countries

• Based on seven base units

• Length: meter (m)

• Mass: kilogram (kg)

• Time: second (s)

• Electrical Current: ampere (A)

• Temperature: kelvin (K)

• Amount of Matter: mole (mol)

• Luminous Intensity: candela (cd)

Derived Units

• Formed by combining base units
• Velocity: $m/s$
• Force: N (newton), where $1 N = 1 kg \cdot m/s^2$
• Energy: J (joule), where $1 J = 1 N \cdot m$

SI Prefixes

• Used to indicate multiples or submultiples of SI units

• yotta (Y): $10^{24}$

• zetta (Z): $10^{21}$

• exa (E): $10^{18}$

• peta (P): $10^{15}$

• tera (T): $10^{12}$

• giga (G): $10^9$

• mega (M): $10^6$

• kilo (k): $10^3$

• hecto (h): $10^2$

• deca (da): $10^1$

• deci (d): $10^{-1}$

• centi (c): $10^{-2}$

• milli (m): $10^{-3}$

• micro ($\mu$): $10^{-6}$

• nano (n): $10^{-9}$

• pico (p): $10^{-12}$

• femto (f): $10^{-15}$

• atto (a): $10^{-18}$

• zepto (z): $10^{-21}$

• yocto (y): $10^{-24}$

• Example: $1 km = 1000 m = 10^3 m$

Unit Conversion

• Process of converting a measurement from one unit to another using conversion factors
• Example: Converting 5 km to meters ($5 km \times \frac{1000 m}{1 km} = 5000 m$)

Significant Figures

• Digits in a number known with certainty plus one estimated digit

Rules for Determining Significant Figures

• Non-zero digits are significant
• Zeros between non-zero digits are significant
• Leading zeros are not significant
• Trailing zeros are significant only if the number contains a decimal point

Examples of Significant Figures

• 1234 has 4 significant figures
• 1002 has 4 significant figures
• 0.0023 has 2 significant figures
• 1.20 has 3 significant figures
• 1200 has 2 significant figures, but 1200. has 4

Rounding

Rounding Rules

• If the next digit is less than 5, the last significant digit remains the same
• If the next digit is 5 or greater, the last significant digit is increased by 1

Examples of Rounding

• Rounding 3.14159 to 3 significant figures: 3.14
• Rounding 1.666 to 2 significant figures: 1.7

Operations with Significant Figures

• Addition and Subtraction: The result should have the same number of decimal places as the number with the fewest decimal places
• Multiplication and Division: The result should have the same number of significant figures as the number with the fewest significant figures

Examples of Operations with Significant Figures

• $1.234 + 2.3 = 3.5$ (rounded from 3.534)
• $4.56 \times 1.2 = 5.5$ (rounded from 5.472)

Vectors

• Quantity with both magnitude and direction, represented by arrows

Vector Components

• A vector $\vec{A}$ can be decomposed into $A_x$ and $A_y$
• $A_x = A \cos(\theta)$
• $A_y = A \sin(\theta)$

Vector Addition

• Add corresponding components, so if $\vec{C} = \vec{A} + \vec{B}$, then:
• $C_x = A_x + B_x$
• $C_y = A_y + B_y$
• $C = \sqrt{C_x^2 + C_y^2}$
• $\theta = \arctan\left(\frac{C_y}{C_x}\right)$

Dot Product (Scalar Product)

• $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta)$
• $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$

Cross Product (Vector Product)

• Results in a vector $\vec{C}$ perpendicular to the plane containing $\vec{A}$ and $\vec{B}$
• $|\vec{C}| = |\vec{A}| |\vec{B}| \sin(\theta)$
• $\vec{C} = \vec{A} \times \vec{B} = (A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x)$

Statistics - Chapter 2 - Random Variables

Basic Definitions

• A random variable $X$ is a function that associates a real number with each possible outcome of a random experiment

Types of Random Variables

• Discrete Random Variable: Variable with a countable set of possible values (finite or countably infinite)
• Continuous Random Variable: Variable that can take any value in a given interval

Discrete Random Variables

Definition

• The probability distribution of a discrete RV $X$ is a function assigning a probability to each possible value of $X$

Mass Function

• $f(x) = P(X = x)$

Conditions

• $0 \leq f(x) \leq 1$
• $\sum_{x} f(x) = 1$

Distribution Function

• Indicates the probability that $X$ takes a value less than or equal to $x$
• $F(x) = P(X \leq x) = \sum_{t \leq x} f(t)$

Mathematical Expectation (Expected Value)

• Weighted average of the possible values of $X$
• $E(X) = \mu = \sum_{x} x f(x)$

Variance

• Measures the dispersion of the values of $X$ around its mean
• $Var(X) = \sigma^2 = E[(X - \mu)^2] = \sum_{x} (x - \mu)^2 f(x) = E(X^2) - [E(X)]^2$

Standard Deviation

• The square root of the variance
• $\sigma = \sqrt{Var(X)}$

Continuous Random Variables

Definition

• The probability distribution is described by a probability density function (pdf) $f(x)$

Density function

• $P(a \leq X \leq b) = \int_{a}^{b} f(x) dx$

Conditions

• $f(x) \geq 0$
• $\int_{-\infty}^{\infty} f(x) dx = 1$

Distribution Function

• Indicates the probability that $X$ takes a value less than or equal to $x$
• $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$

Mathematical Expectation (Expected Value)

• Average weighted values with weights given by pdf
• $E(X) = \mu = \int_{-\infty}^{\infty} x f(x) dx$

Variance

• Measures the dispersion around the mean
• $Var(X) = \sigma^2 = E[(X - \mu)^2] = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx = E(X^2) - [E(X)]^2$

Standard Deviation

• Square root of variance
• $\sigma = \sqrt{Var(X)}$

Common Distributions

Discrete Uniform Law

• All possible values ​​of $X$ have the same probability
• $P(X = x) = \frac{1}{n}$

Bernoulli's Law

• Only has 2 values, 0(failure) and 1(success)
• $P(X = 1) = p$
• $P(X = 0) = 1 - p$

Binomial Law

• Represents the number of successes in n independent Bernoulli trials
• $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

Poisson's Law

• Represents the number of events that occur in a given interval of time or space
• $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

Exponential Law

• Represents the waiting time until an event occurs
• $f(x) = \lambda e^{-\lambda x}$

Normal Law

• Symmetrical bell shaped distribution
• $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$