Patterns

🎯 Statistics 1: Pattern Recognition Guide (V5)

This guide dissects every major question pattern encountered in Stats 1 assignments, mocks, and exams. Master these “recipes” to solve any variation with clinical precision.

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📦 Volume I: Descriptive & Bivariate Patterns

Pattern 1: The “Correction” Triage (Mean & Variance)

• Problem: You are given a mean/variance, then told one or more values were incorrectly recorded.
• The Master Recipe:
  1. Correct the Sum: $\sum x_{new}=\sum x_{old}-x_{wrong}+x_{correct}$.
  2. Correct the Mean: ${\bar{x}}_{new}=\frac{\sum x_{new}}{n}$.
  3. Correct the Squared Sum: $\sum x_{new}^2=\sum x_{old}^2-x_{wrong}^2+x_{correct}^2$.
     • Note: Find $\sum x_{old}^2$ using $Var(X)=\frac{\sum x^2}{n}-(\bar{x}{)}^2$.
  4. Re-calculate Variance: Using $\sum x_{new}^2$ and ${\bar{x}}_{new}$.

Pattern 2: The “Scale & Shift” Transformation

• Problem: Find the new mean/variance/r after adding a constant $c$ or multiplying by $k$.
• The Master Recipe:
  • $E[kX+c]=kE[X]+c$
  • $Var(kX+c)=k^{2}Var(X)$
  • $SD(kX+c)=|k|SD(X)$
  • $r(kX+c,mY+d)=r(X,Y)$ if $k,m$ have the same sign; else $-r(X,Y)$.

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🎲 Volume II: Counting & Probability Patterns

Pattern 3: The “License Plate / Arrangement” Logic

• Problem: Forming strings/codes with specific constraints (e.g., vowels together, digits can’t repeat).
• The Master Recipe:
  • Slot Method: Draw slots for each position and fill in the number of choices.
  • Bundle Method: If items must stay together, treat them as one single “super item,” then multiply by the internal arrangements of that bundle.

Pattern 4: The Bayes Theorem “Evidence Update”

• Problem: Given a test result (positive/negative), find the probability of the true state (diseased/healthy).
• The Master Recipe (The Table Method): | State | Prior $P(B_i)$ | Likelihood $P(A|B_i)$ | Product | | :--- | :--- | :--- | :--- | | Disease | $p$ | Sensitivity | $p\times Sen$ | | Healthy | $1-p$ | 1 - Specificity | $(1-p)\times (1-Spec)$ |
  • Total Prob ($P(A)$): Sum of Product column.
  • Final Answer: $\frac{Product(TargetState)}{TotalProb}$.

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📈 Volume III: Random Variable Patterns

Pattern 5: The “PMF Constant” Rescue

• Problem: You are given a PMF $f(x)=k\cdot g(x)$ and asked to find $k$.
• The Master Recipe:
  1. Axiom: $\sum f(x)=1$.
  2. Execution: $k\cdot \sum g(x)=1⟹k=\frac{1}{\sum g(x)}$.
  3. Trap: Ensure you only sum over the Support of $X$ (The values where $g(x)\ne 0$).

Pattern 6: The “Linearity Shortcut”

• Problem: Find $E[X_1+X_2+\cdots +X_n]$ for complex dependent variables.
• The Master Recipe:
  • Magic Property: $E[\sum X_i]=\sum E[X_i]$.
  • Execution: Calculate the mean of one trial/item, then multiply by $n$. You do not need to find the joint distribution.

Pattern 7: The “Poisson Scaling” Identity

• Problem: Given $\lambda =3$ per hour, find $P(X=0)$ in 20 minutes.
• The Master Recipe:
  1. Normalize Time: $20\text{ mins}=1/3\text{ hour}$.
  2. Scale Rate: ${\lambda }_{new}={\lambda }_{old}\cdot \text{scale}=3\cdot (1/3)=1$.
  3. Calculate: $P(X=0)=\frac{e^{-1}{1}^0}{0!}=e^{-1}\approx 0.368$.

Pattern 8: The “Binomial vs Poisson” Triage

• Problem: Which distribution to use?
• The Selection Recipe:
  • Binomial: If $n$ is small and fixed, and $p$ is significant (e.g., $n=10,p=0.4$).
  • Poisson: If $n$ is very large (or infinite) and we focus on the rate of occurrence (e.g., “3 cars per min”).
  • Approximation: If $n\ge 100$ and $np\le 5$, use Poisson for simpler calculation.

Pattern 9: The “Memoryless” Survival Hack

• Problem: Exponential Dist. $P(X>10|X>5)$.
• The Master Recipe:
  • Logic: The variable “forgets” it survived 5 hours.
  • Identity: $P(X>s+t|X>s)=P(X>t)=e^{-\lambda t}$.
  • Execution: Here $s=5,t=5$. Ans: $P(X>5)=e^{-5\lambda }$.

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🏆 Mastermind Exam Triage

When faced with a complex problem, apply this filter:

1. Is it a count? $\to$ Binomial or Poisson.
2. Is it a measurement? $\to$ Uniform or Exponential.
3. Are there multiple states? $\to$ Bayes Theorem.
4. Are there ‘wrong’ values? $\to$ Mean/Var Correction Pattern.