Mathematical Methods in PDEs and Statistical Analysis
Partial Differential Equations (PDE)
A Partial Differential Equation (PDE) involves a function u(x, y, …) and its partial derivatives.
- Homogeneous: If every term in the equation contains the dependent variable u or its derivatives. The general solution is simply the Complementary Function (C.F.).
- Non-Homogeneous: If there is a term that is a function of the independent variables only (f(x, y)). The solution is u = C.F. + P.I. (Particular Integral). Example: ∇2u = f(x, y) (Poisson’s Equation).
Fundamentals of Statistics: Concepts and Data Analysis
1. Statistics: Descriptive vs. Inferential
Statistics is the science of collecting, organizing, presenting, analyzing, and interpreting data to draw meaningful conclusions and support decision-making.
Comparison of Statistical Methods
| Basis | Descriptive Statistics | Inferential Statistics |
|---|---|---|
| Meaning | Summarizes and describes data | Draws conclusions about population |
| Purpose | To present data clearly | To make predictions/decisions |
| Data used | Uses complete data set | Uses sample data |
| Techniques | Mean, median, mode, graphs | Probability, |
Maximum unambiguous range maximum theoretical range
Interquartile range = range between the first and third quartile.
Cumulative frequency = sum of all frequencies for all values.
Variance = the average of the squared differences from the Mean.
Standard variation = the average of the squared differences from the Mean under a squared root (the same as Variance just under a square root to get rid of the squared unit).
The Range = The distance between two values of which we combine their frequencies to simplify longer datasets.
Quartiles = A division of
Read MoreSOC 222: Measuring the Social World Study Notes
SOC 222: Measuring the Social World
Key Concepts and Definitions
Population vs. Sample
- Population: The entire group you want to study. Example: All students at UTM.
- Sample: A subset of the population used to make conclusions. Example: 100 UTM students surveyed in the library.
- Population Parameter: The true value in the population. Example: The actual percentage of all UTM students who cheat.
- Sample Statistic: The estimate derived from the sample. Example: 15% of surveyed students admit to cheating.
- Sampling
Statistical Analysis: Regression and Probability Models
Regression Analysis and Predictive Modeling
Regression analysis is a statistical method used to model the relationship between variables and to predict the value of one variable using another.
Main Types of Regression
- Simple linear regression: One independent variable and one dependent variable.
- Multiple regression: Several independent variables predicting one dependent variable.
- Logistic regression: Used when the dependent variable represents probabilities or categories.
The goal of simple linear regression
Read MoreEssential Statistics: Sampling, Distributions, and Testing
1. Sampling and Basic Concepts
Population: The entire group being studied.
Sample: A subset of the population.
Example
- Population: All university students.
- Sample: 200 students surveyed.
Parameter vs. Statistic
- Parameter: A numerical value describing a population.
- Statistic: A numerical value derived from a sample.
Examples:
- p = True population proportion.
- p̂ (p-hat) = Sample proportion.
Sample Proportion Formula
p̂ = x / n
Where:
- x = Number of successes.
- n = Sample size.
Example: 48 support a policy out of 80.
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