Module 3

Comparing Complex Numbers - Algebraic Form

• Given $z=a+ib$ and $z'=a'+ib'$, where $a, a', b, b'$ are real numbers, $z=z'$ if and only if $a=a'$ and $b=b'$.

Comparing Complex Numbers - Modulus

• If $|z| = |z'|$, it does not necessarily mean that $z=z'$.
• Example: While $|3+4i| = |5| = 5$, but $3+4i$ isn't equal to $5$.

Comparing Complex Numbers - Argument

• If two non-zero complex numbers have the same argument ($arg(z) = arg(z')$), it does not automatically mean that $z=z'$.
• Example: $arg(1+i) = arg(2+2i) = \frac{\pi}{4}$, but $1+i$ is not the same as $2+2i$.

Conclusion Regarding Order in Complex Numbers

• The standard "less than or equal to" order relationship ($\le$) is not defined within the set of complex numbers ($\mathbb{C}$).
• While an order relation could be defined, it would lack compatibility with standard mathematical operations.

Economics Definition

• Economics studies how societies use scarce resources to produce valuable goods and services and distribute them among individuals.

Key Economic Concepts

• Goods are scarce, indicating limited availability.
• Society needs to employ resources efficiently.

Branches of Economics

• Microeconomics focuses on the behavior of individual entities like markets, firms, and households.
• Macroeconomics examines the overall performance of the economy.

Economic Tools - Scientific Approach

• Examining economic questions draws on statistics and historical data.

Economic Reasoning - Common Errors

• The post hoc fallacy involves assuming causation from a sequence of events.
• Failure to keep all other factors constant.
• The fallacy of composition assumes what is true for a part is also true for the whole.

Production - Inputs and Outputs

• Inputs (Factors of production): Goods or services used to produce other goods or services (land, labor, and capital).
• Products (Outputs): The resulting goods or services from the production process.

The Production Possibility Frontier (PPF)

• A curve showing the maximum amounts of production attainable by an economy, given its technological knowledge and available inputs.

PPF - Example

• Shows the trade-off between producing food and computers:
  • 0 food = 15 computers
  • 1 food = 14 computers
  • 2 food = 12 computers
  • 3 food = 9 computers
  • 4 food = 5 computers
  • 5 food = 0 computers

Cost of Opportunity

• The cost of the forgone alternative when making a decision.

Productive Efficiency

• Achieved when an economy cannot produce more of one good without producing less of another, placing it on the PPF.

Atmospheric Pollution - Acid Rain

• Acid rain forms from pollutants like sulfur oxides (SOx) and nitrogen oxides (NOx) that react in the atmosphere to form acids, subsequently deposited via rain, snow, or fog.

Consequences of Acid Rain

• Acidification harms aquatic life and agriculture.
• Structures and monuments undergo corrosion.
• Vegetation, forests, and crops sustain damage.
• Respiratory diseases and other health complications arise.

Atmospheric Pollution - Ozone Layer Depletion

• The ozone layer contains a high concentration of ozone (O3), which absorbs much of the sun's UV radiation. Chlorofluorocarbons (CFCs), halons, and other compounds diminish the ozone concentration.

Consequences of Ozone Layer Depletion

• Increased UV radiation leads to skin cancer, cataracts, and compromised immune systems.
• Marine ecosystems, particularly phytoplankton and fish larvae, are harmed.
• Crop productivity is reduced.

Atmospheric Pollution- Greenhouse Effect

• The greenhouse effect is a natural occurrence where gases like carbon dioxide (CO2), methane (CH4), and nitrous oxide (N2O) trap heat, maintaining suitable temperatures. Increased gas concentrations, primarily from burning fossil fuels, deforestation, and agriculture, intensify the effect.

Consequences of the Greenhouse Effect

• The planet's average temperature is rising, causing more frequent and intense heatwaves.
• Melting of glaciers and polar ice caps results in rising sea levels.
• Altered precipitation patterns including droughts affecting water supply and agriculture.
• Extreme weather events like hurricanes, storms, and floods increase in frequency and intensity.

Water Pollution - Eutrophication

• Eutrophication results from an excess of nutrients, like nitrogen and phosphorus, in water bodies. The overabundance causes excessive growth of algae and aquatic plants, depleting the water's oxygen.

Consequences of Eutrophication

• Lowered oxygen levels in water bodies lead to the death of fish and other aquatic organisms.
• Potentially hazardous algal blooms are likely to proliferate, contaminating water and endangering human health.
• Aquatic biodiversity declines.
• The water is less suitable for human consumption, irrigation, or other operations.

Water Pollution - Heavy Metal Contamination

• Heavy metal contamination occurs when toxins such as mercury, lead, and cadmium pollute water bodies, contaminating water and sediments. Metals can be bioaccumulated by aquatic organisms.

Consequences of Heavy Metal Contamination

• Heavy metal contamination leads to neurological, kidney, and liver damage.
• Foods become unfit for human consumption.
• Aquatic organisms can suffer, changing behavior, reproduction, and development.

Soil Pollution - Agrochemical Contamination

• Occurs when agrochemicals are improperly used in agriculture, contaminating soil and water. Humans can be subject to pesticides via food and water.

Consequences of Agrochemical Contamination

• Health conditions like cancer, congenital defects, and neurological issues can be the result of agrochemicals.
• Fruits, vegetables, and legumes become unfit for consumption.
• Ecological equilibrium may be altered, allowing the establishment of pesticide-resistant pests.
• Fauna and flora can be harmed.

Soil Pollution - Inadequate Solid Waste Disposal

• Improper disposal of municipal, industrial, and hospital waste pollutes soil and water.

Consequences of Inadequate Solid Waste Disposal

• Soil and water acquire toxins present in wastes.
• Odors emerge and visuals are compromised.
• Disease vectors like rats, cockroaches, and mosquitoes will thrive.
• Landfills are subject to explosions and fires.

Soil Pollution - Leaching

• Leaching is the process where rainwater penetrates the soil and drags substances from solid waste, such as heavy metals and organic compounds, contaminating the soil and groundwater.

Consequences of Leaching

• Water and soil become toxic.
• Water use becomes limited.
• Flora and fauna will suffer.

Pollution Summary - Atmospheric Pollution

• Acid Rain: SOx and NOx lead to acidification, corrosion, vegetation damage, and health problems.
• Ozone Layer Destruction: CFCs, halons increase UV radiation and harm the marine environment.
• Greenhouse Effect: CO2, CH4, N2O causing increased temperatures and extreme climate conditions.

Pollution Summary - Water Pollution

• Eutrophication: Excess nutrients lead to decreased oxygen and toxic algae.
• Heavy Metals: Hg, Pb, Cd leading to health problems, and contaminated food.

Pollution Summary - Soil Pollution

• Agrochemicals: Inadequate employment leads to health problems and an unbalanced ecological system.
• Improper Waste Disposal leads to soil water contamination and vermin.
• Leaching contaminates water and harms fauna and flora.

Statistical Inference - Point Estimation Definition

• A point estimator is a statistic used to estimate a population parameter.

Statistical Inference - Likelihood Function Definition

• Given a random sample $W$ from a population with pdf $f(x;\theta)$, where $\theta$ is an unknown parameter, the likelihood function is: $L(\theta;x_1,\dots,x_n) = \prod_{i=1}^{n} f(x_i;\theta)$

Statistical Inference - Unbiased Estimator Definition

• $\hat{\theta}$ is an unbiased estimator of $\theta$ if $E(\hat{\theta}) = \theta$.
• Otherwise, $\hat{\theta}$ is biased.
• The bias is given by $Bias(\hat{\theta}) = E(\hat{\theta}) - \theta$.

Statistical Inference - Efficiency of Estimators Definition

• $\hat{\theta}_1$ is more efficient than $\hat{\theta}_2$ if $Var(\hat{\theta}_1) < Var(\hat{\theta}_2)$, where $\hat{\theta}_1$ and $\hat{\theta}_2$ are unbiased estimators of $\theta$.

Statistical Inference - Mean Squared Error (MSE) Definition

• The MSE of $\hat{\theta}$ is given by: $MSE(\hat{\theta}) = E[(\hat{\theta} - \theta)^2] = Var(\hat{\theta}) + [Bias(\hat{\theta})]^2$

Statistical Inference - Consistent Estimator Definition

• $\hat{\theta}$ is a consistent estimator of $\theta$ if $\hat{\theta}$ converges to $\theta$ in probability, meaning for any $\epsilon > 0$: $\lim_{n \to \infty} P(|\hat{\theta} - \theta| < \epsilon) = 1$

Statistical Inference - Sufficient Statistic Definition

• $\hat{\theta}$ is a sufficient statistic for $\theta$ if it contains all information about $\theta$ in the sample such that the conditional distribution of the sample given $\hat{\theta}$ does not depend on $\theta$.

Statistical Inference - Confidence Intervals Definition

• A confidence interval is an interval to estimate a population parameter; the confidence level is the probability the interval contains the true value.

Statistical Inference - Confidence Interval for $\mu$ (Normal Population)

• For a random sample $W$ from a normal population with mean $\mu$ and variance $\sigma^2$, a $100(1 - \alpha)%$ confidence interval for $\mu$ is: $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
  • Where $\bar{x}$ is the sample mean.
  • $s$ is the sample standard deviation.
  • $t_{\alpha/2, n-1}$ is the t-value.

Statistical Inference - Confidence Interval for $\sigma^2$ (Normal Population)

• For a random sample $W$ from a normal population with mean $\mu$ and variance $\sigma^2$, a $100(1 - \alpha)%$ confidence interval for $\sigma^2$ is: $(\frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}})$
  • Where $s^2$ is the sample variance.
  • $\chi^2_{\alpha/2, n-1}$ is the $\chi^2$-value.

Statistical Inference - Hypothesis Testing Definition

• A hypothesis test is a procedure for testing a claim about a population parameter.

Statistical Inference - Null and Alternative Hypothesis Definitions

• The null hypothesis is the claim being tested.
• The alternative hypothesis is the claim being supported.

Statistical Inference - Type I and Type II Error Definitions

• A type I error is rejecting the null hypothesis when it is true, with probability $\alpha$.
• A type II error is failing to reject the null hypothesis when it is false, with probability $\beta$.

Statistical Inference - Power of a Test Definition

• The power of a test is the probability of rejecting the null hypothesis when it is false, given by $1 - \beta$.

Statistical Inference - P-Value Definition

• The p-value is the probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

Statistical Inference - Hypothesis Test for $\mu$ (Normal Population)

• To test $H_0: \mu = \mu_0$ versus $H_1: \mu \neq \mu_0$ for a random sample $W$ from a normal population, use the test statistic: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$
  • Where $\bar{x}$ is the sample mean
  • $s$ is the sample standard deviation.
  • $n$ is the sample size. The test statistic has a $t$ distribution with $n - 1$ degrees of freedom.

Statistical Inference - Hypothesis Test for $\sigma^2$ (Normal Population)

• To test $H_0: \sigma^2 = \sigma^2_0$ versus $H_1: \sigma^2 \neq \sigma^2_0$ for a random sample $W$ from a normal population, use the test statistic: $\chi^2 = \frac{(n-1)s^2}{\sigma^2_0}$
  • Where $s^2$ is the sample variance.
  • $n$ is the sample size. The test statistic has a $\chi^2$ distribution with $n - 1$ degrees of freedom.

Density (ρ) Definition

• Density (ρ) is a substance's mass per unit volume. $\qquad \rho = \frac{m}{V}$

Standard Unit For Density

• Density is measured in kilograms per cubic meter ($\frac{kg}{m^3}$) in the SI system.

Density is Intensive

• Density is an intensive property, meaning the value is independent of the sample size.

Density Dependency on Temperature

• Densities change along with temperature. Air for example decreases in density as temperature increases.

Density Example

• Lead is much denser than wood, ships of lead float based on their large volume and large amount of water displacement.

Chemical Principles - Density Defined

• Density relates a substance's mass to the amount of space, and is mathematically defined by: $\qquad Density = \frac{Mass}{Volume}$ or $\rho = \frac{m}{V}$ $\qquad \rho$: density $\qquad m$: mass $\qquad V$: volume

Units of Density

• Solids and liquids have density measure in grams per/cubic centimeter (g/cm³) or kilograms/cubic meter (kg/m³).
• Gases' density are measured in grams per liter (g/L) or kilograms/cubic meter (kg/m³).

Temperature Dependence

• Density is dependent on temperature. As temperature increases, most substances expand.

Density Example

• A metal cube is 500g and has a volume of 200cm³ (density = 2.5 g/cm³): $\qquad Density = \frac{500g}{200cm³} = 2.5 g/cm³$

States of Matter

• Matter is present in many states with distinct physical properties - solid, liquid and gas.

Matter Explained - Solids

• Solids are tightly packed in a fixed arrangement. Solids have consistent shape and volume.

Matter Explained - Liquids

• Closely but moveable particles allow liquids to take shape and become almost incompressible.

Matter Explained - Gases

• Gases are extremely dispersed and travel in random. Gases compress and take any shape or form.

Phase Transitions - List

• The following list describes the phase transitions:
  • Solid to Liquid Melting
  • Liquid to Solid Freezing
  • Liquid to Gas Boiling
  • Gas to Liquid Condensation
  • Solid to Gas Sublimation
  • Gas to Solid Deposition