Stats1_Prerequisites_Comprehensive

Stats 1: Comprehensive Prerequisites Guide

IMPORTANT

Goal: This guide covers the exact mathematical tools you need to survive Stats 1. If you are stuck on a problem, 90% of the time it’s because of one of these concepts.

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1. Basic Arithmetic & Algebra

1.1 Summation Notation ($\sum$)

• Symbol: The Greek letter Sigma ($\sum$) means “Add them all up”.
• Formula: ${\sum }_{i=1}^n{x}_i=x_1+x_2+\cdots +x_n$.
• Properties:
  1. $\sum (x_i+y_i)=\sum x_i+\sum y_i$ (You can split sums).
  2. $\sum (c\cdot x_i)=c\cdot \sum x_i$ (Constants move outside).
  3. ${\sum }_{i=1}^{n}c=n\cdot c$ (Adding a constant $n$ times).
  • Trap: $\sum (x_i\cdot y_i)\ne (\sum x_i)\cdot (\sum y_i)$. Never do this.

1.2 Linear Equations ($y=mx+c$)

• Slope ($m$): How much $y$ changes when $x$ increases by 1.
  • $m>0$: Positive relationship (Both go up).
  • $m<0$: Negative relationship (One up, one down).
• Intercept ($c$): The value of $y$ when $x=0$.

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2. Set Theory (The Language of Probability)

2.1 Operations

1. Union ($\cup$): “OR”. Everything in A plus everything in B.
   • $A=\{1,2\},B=\{2,3\}⟹A\cup B=\{1,2,3\}$.
2. Intersection ($\cap$): “AND”. Only what is in BOTH.
   • $A\cap B=\{2\}$.
3. Complement ($A^c$ or $A^{\prime }$): “NOT”. Everything outside A.
   • If Universe $S=\{1,2,3,4\}$ and $A=\{1,2\}$, then $A^c=\{3,4\}$.

2.2 Venn Diagrams

• Mutually Exclusive: Two circles that never touch. ($A\cap B=\varnothing$).
• Subset: Circle A is completely inside Circle B. ($A\subset B$).

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3. Combinatorics (Counting)

3.1 Factorial ($n!$)

• $n!=n\times (n-1)\times \cdots \times 1$.
• $0!=1$ (Memorize this!).
• $5!=120$.

3.2 Permutation vs Combination

Feature	Permutation (${}^n{P}_r$)	Combination (${}^n{C}_r$)
Keywords	Arrange, Rank, Order, Line up	Select, Group, Team, Choose
Order?	Matters (AB $\ne$ BA)	Doesn’t Matter (AB = BA)
Formula	$\frac{n!}{(n-r)!}$	$\frac{n!}{r!(n-r)!}$

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4. Calculus (For Continuous Random Variables)

• Note: Stats 1 is mostly discrete, but Week 8 touches on continuous distributions (Uniform/Normal).
• Area Under Curve: Probability = Area.
  • Total Area under PDF = 1.
  • $P(a<X<b)=\text{Area between }a\text{ and }b$.

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5. Common Pitfalls

1. Percent vs Probability:
   • $50\%=0.5$. Always convert to decimal for calculations.
2. “At Least One”:
   • Always calculate $1-P(\text{None})$. It’s faster.
3. Variance Units:
   • Variance is in ${\text{units}}^2$. Standard Deviation is in $\text{units}$.
   • Always take $\sqrt{\text{Variance}}$ to get SD before interpreting.