Advanced Mathematics and Statistics Reference
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)
Read MoreAdvanced Vector Calculus and Matrix Algebra Problems
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(
Read MoreAlgebra and Calculus Problem Solutions
Mathematical Problem Set 1
- Profit Function: P(x) = -4x2 + 632x – 4000, where x > 10 and x is the number of units produced. Find the total production for $800 profit.
Work: 0 = -4x2 + 632x – 4800. Plug into the quadratic formula.
ANS: x = 150 - Relative Maximum: F(x) = ln(2x – 8.4) – 0.25(x + 4)2 + 9.83.
ANS: 10.2 at x = 5.5 - Real Zeros: 3 – 2 log5 √(x2 + 2x + 3), to three decimal places.
ANS: x = -12.091, x = 10.091 - Carbon-14 Dating: A 2,500-year-old estimate of Carbon-14 shows a decrease from 20 grams
Functions & Algebra: Domain, Graphing, and Key Techniques
Function Basics
Definition: A function assigns exactly one output for each input.
Function notation: f(x)
Vertical Line Test: If any vertical line intersects a graph more than once, the relation is not a function.
Using Tables to Graph (by Hand)
Steps:
- Choose x-values (use symmetry if possible).
- Plug the x-values into the function.
- Create a table of (x, f(x)).
- Plot the points.
- Connect points smoothly to represent the graph.
Always check:
- x-intercepts
- y-intercept
- asymptotes (if any)
Domain Rules (Very Important)
Read MoreExcel Import, Cleaning, Transformation & Visualization
Importing and Exporting Data
In data science, importing is the act of bringing external data into Excel for analysis, while exporting is saving your Excel data into a format that other programs can read.
Common File Formats
Understanding these formats helps you choose the right one for your task:
- Excel Workbook (.xlsx): A workbook that stores data, formulas, formatting, charts, and multiple sheets. Best for your working file.
- CSV (Comma-Separated Values): A plain text file where a comma separates each