Probability and Statistics: Key Concepts
Sample Space (Ω)
The collection of all possible outcomes of an experiment.
Empty Set (∅)
Indicates that an event E is impossible.
Empirical Probability (EP)
- What is observed.
- Data collection.
- Estimation of some unknown truth.
Theoretical Probability (TP)
What is expected – calculated using mathematical reasoning or computation.
Law of Large Numbers
EP approaches TP as n approaches infinity.
Union of Joint Events (A ∪ B)
A ∪ B: Everything is shaded (“A or B”).
Joint: if they intersect or overlap.
Union of
Read MoreUnderstanding Data Dispersion: Standard Deviation & More
Understanding Data Dispersion
How can we tell if data is dispersed or not?
- There is no clear interpretation of the standard deviation. It depends on the context and our criterion.
- For example, in a sample of screws for technical equipment, where we measure the size of each screw, we expect a dispersion close to zero. Otherwise, it will mean we have to consider lots of screws as defective.
- On the other hand, on a sample of workers in a company and their salaries, we should expect high dispersion in the
Solving First-Order Ordinary Differential Equations
Separable First-Order ODEs
- Separate the variables by putting all terms with “y” on one side of the equation and all terms with “x” on the other side.
- Integrate both sides of the equation with respect to their respective variables.
- Solve for y to find the general solution, adding a constant of integration at the end of the integration step; if an initial condition is given, substitute it to find the particular solution.
Key Steps in Detail
A separable first-order ODE can be written as dy/dx = f(x)g(y)
Read MoreDecision-Making Strategies: Risk, Certainty, and Uncertainty
Cooperatives grant members several rights, including participation in activities, access to necessary information, fulfillment of obligations, and the return of cooperative surpluses.
The Decision Matrix
A decision matrix is a table that encompasses all elements involved in decision-making. It provides a starting point for solving problems and offers a structured analysis:
- Strategies: Controllable variables representing alternatives or options to choose from.
- States of Nature: Uncontrollable variables
Understanding Quantitative Data Collection in Ecology
Quantitative data collection in the ecology of a habitat, community, or town involves interpreting collected information to draw conclusions.
Population statistics encompass all the data from which we wish to draw conclusions.
Statistical sample: A statistical sample is a randomly selected subset of the statistical population.
Statistics can be divided into descriptive analysis and statistical inference.
Descriptive analysis: This involves comments, graphics, tabular data, or numerical data that allow
Read MoreSolving Mixture and Differential Equations: Step-by-Step
Mixture Problems and Differential Equations
This document presents solutions to two types of mathematical problems: mixture problems and differential equations. The solutions are presented step-by-step, with explanations.
2.3.35 Mixture Problems
A tank contains 500L of water with 0.2 kg of salt. Initially, there are 5 kg of salt in the tank. The inflow and outflow rates are 5 L/min. We will determine the concentration at 10 minutes and after a leak develops, at 20 minutes.
a) Concentration at 10 minutes
Let
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