Discrete Mathematics: Sets, Graphs, and Matrix Algebra
Unit 1: Set Theory and Functions
1. Set
A set is a well-defined collection of distinct objects, called elements or members. These objects can be numbers, symbols, people, or even other sets. Sets are usually denoted by capital letters such as A, B, or C, and their elements are written within curly braces. For example, A = {1, 2, 3}. A set must be well-defined, meaning it should be clear whether a given object belongs to the set or not. Sets play a foundational role in mathematics and computer science
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Business Statistics: Data-Driven Insights
Business statistics are vital for informed decision-making, performance evaluation, and risk management. They utilize data analysis to provide insights into market trends, customer behavior, and operational efficiency. The scope is broad, covering areas like forecasting, production planning, marketing analysis, and financial modeling, helping businesses make strategic plans based on data rather than intuition.
Importance of Business Statistics
- Informed Decision-
Fundamentals of Statistical Analysis and Data Collection
Nature of Statistics
The fundamental question regarding Statistics is whether it is a science or an art. Professor Tippet rightly observed that “Statistics is both a science as well as an art.”
- As a science, Statistics studies numerical data in a scientific or systematic manner.
- As an art, Statistics relates quantitative data to real-life problems.
By using statistical data, we are able to analyse and understand real-life problems much better than otherwise. Thus, the problem of unemployment in India
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Probability Density Functions and Distributions
A function f(x) is a valid probability density function (PDF) if:
- f(x) ≥ 0 for all values of x.
- ∫-∞∞ f(x)dx = 1.
Expectation and Variance
- Expected value (mean): E[X] = ∫-∞∞ x ⋅ f(x)dx
- Expected value of X2: E[X2] = ∫-∞∞ x2 ⋅ f(x)dx
- Variance: Var(X) = E[X2] – (E[X])2
Special Probability Models
- Uniform Distribution: f(x) = 1/(b-a), for a ≤ x ≤ b.
- Uniform Distribution Expected Value: (a + b)/2
- Exponential Distribution: f(x) = λe-λx,
Atmosphere and Weather Exam Cheat Sheet — Key Formulas & Tips
Atmosphere & Weather Exam Cheat Sheet (Super Simple)
Exam-Style Questions — How to Answer (with steps)
1. Temperature Questions
Q: Convert Celsius ↔ Kelvin
How to answer:
- If going from °C → K: add 273.15 (K = °C + 273.15).
- If going from K → °C: subtract 273.15 (°C = K − 273.15).
2. Insolation / Energy Questions
Q: Why is this day warmer/colder?
How to answer:
- Look at the irradiance graph.
- Higher irradiance = more sunlight.
- More sunlight usually = warmer temperatures.
3. Relative Humidity (RH)
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Cayley-Hamilton Theorem: Verification and Inverse
Problem 1: Verify Cayley-Hamilton theorem for A and find A-1
Given matrix: A = [[2, 1, 1], [0, 1, 0], [1, 1, 2]]
Step 1: Find the Characteristic Equation |A – λI| = 0
A – λI = [[2-λ, 1, 1], [0, 1-λ, 0], [1, 1, 2-λ]]
Expand along the 2nd row (which has two zeros):
|A – λI| = (1-λ) × |[2-λ, 1], [1, 2-λ]|
= (1-λ) × [(2-λ)(2-λ) – 1×1]
= (1-λ) × [(2-λ)2 – 1]
= (1-λ) × [4 – 4λ + λ2 – 1]
= (1-λ)(λ2 – 4λ + 3)
= (1-λ)(λ-1)(λ-3)
= -(λ-1)2(
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