1. Descriptive Statistics
Measures of Central Tendency
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Mean (Arithmetic Average)
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Raw data: $$\displaystyle \bar{x} = \frac{\sum x_i}{n} $$
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Grouped data: $$\displaystyle \bar{x} = \frac{\sum f_i x_i}{\sum f_i} $$ (where $$\displaystyle x_i $$ = class midpoint)
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Merits: Uses all data, mathematically tractable. Demerits: Affected by extremes, not for open-ended classes.
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Median
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Position: $$\displaystyle \left(\frac{n+1}{2}\right)^{th} $$ value (raw) or $$\displaystyle \frac{N}{2}^{th} $$ cumulative frequency (grouped).
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Formula (grouped): $$\displaystyle M = L + \left(\frac{\frac{N}{2} - C_f}{f}\right) \times h $$
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Merits: Robust to outliers. Demerits: Not for algebraic manipulation, ignores extreme values.
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Mode
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Raw: Most frequent value.
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Grouped: $$\displaystyle M_o = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h $$
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Merits: Actual most common value. Demerits: May be multiple/undefined, unstable.
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[!TIP] For grouped data, ensure classes are continuous and of equal width for median/mode formulas.
Measures of Dispersion
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Range: $$\displaystyle R = \max(x_i) - \min(x_i) $$. Simple but ignores distribution.
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Variance & Standard Deviation
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Population: $$\displaystyle \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} $$, $$\displaystyle \sigma = \sqrt{\sigma^2} $$
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Sample: $$\displaystyle s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} $$, $$\displaystyle s = \sqrt{s^2} $$
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Shortcut: $$\displaystyle \sigma^2 = \frac{\sum x_i^2}{N} - \mu^2 $$
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Mean Deviation (MD)
- $$\displaystyle MD = \frac{\sum |x_i - \text{median}|}{n} $$ (or mean). Less sensitive but mathematically awkward.
Measures of Shape
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Skewness (asymmetry)
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Pearson’s: $$\displaystyle Sk_p = \frac{\bar{x} - \text{Mode}}{s} $$ or $$\displaystyle \frac{3(\bar{x} - M)}{s} $$
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Moment-based: $$\displaystyle \gamma_1 = \frac{\mu_3}{\sigma^3} $$ (zero for symmetric).
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Kurtosis (peakedness)
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β₂ (Pearson’s): $$\displaystyle \beta_2 = \frac{\mu_4}{\sigma^4} $$. Normal: β₂ = 3.
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Excess kurtosis: $$\displaystyle \gamma_2 = \beta_2 - 3 $$. Leptokurtic (>0), Platykurtic (<0).
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2. Fundamentals of Probability
Basic Concepts
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Experiment: Process with uncertain outcome.
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Sample Space (S): Set of all possible outcomes.
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Event (E): Subset of $S$.
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Axioms: $0 \le P(E) \le 1$; $$\displaystyle P(S)=1 $$; For mutually exclusive $$\displaystyle E_i $$, $$\displaystyle P(\cup E_i) = \sum P(E_i) $$.
Conditional Probability & Independence
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$$\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)} $$, $$\displaystyle P(B) > 0 $$.
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Independence: $$\displaystyle P(A \cap B) = P(A)P(B) $$.
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Multiplication Rule: $$\displaystyle P(A \cap B) = P(A)P(B|A) = P(B)P(A|B) $$.
Bayes’ Theorem
For partitions $$\displaystyle B_1, B_2, \dots, B_k $$:
$$P(B_i|A) = \frac{P(B_i)P(A|B_i)}{\sum_{j=1}^k P(B_j)P(A|B_j)}$$
[!TIP] Common in medical diagnosis and quality control (e.g., defective items from different sources).
Counting Techniques
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Permutations (ordered): $$\displaystyle ^nP_r = \frac{n!}{(n-r)!} $$
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Combinations (unordered): $$\displaystyle ^nC_r = \frac{n!}{r!(n-r)!} $$
3. Random Variables
Types
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Discrete: Countable values (e.g., number of heads).
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Continuous: Any value in an interval (e.g., height).
Probability Functions
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PMF (discrete): $$\displaystyle p(x) = P(X=x) $$, $$\displaystyle \sum p(x) = 1 $$.
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PDF (continuous): $f(x) \ge 0$, $$\displaystyle \int_{-\infty}^{\infty} f(x)dx = 1 $$, $$\displaystyle P(a<X<b) = \int_a^b f(x)dx $$.
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CDF: $$\displaystyle F(x) = P(X \le x) $$. Properties: non-decreasing, $$\displaystyle \lim_{x \to -\infty}F(x)=0 $$, $$\displaystyle \lim_{x \to \infty}F(x)=1 $$.
Expectation & Variance
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Expectation: $$\displaystyle E(X) = \sum x p(x) $$ (discrete), $$\displaystyle \int x f(x)dx $$ (continuous).
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Variance: $$\displaystyle V(X) = E[(X - E(X))^2] = E(X^2) - [E(X)]^2 $$.
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Properties:
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$$\displaystyle E(aX + b) = aE(X) + b $$
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$$\displaystyle V(aX + b) = a^2 V(X) $$
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$$\displaystyle E\left(\sum X_i\right) = \sum E(X_i) $$ (no independence needed)
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$$\displaystyle V\left(\sum X_i\right) = \sum V(X_i) $$ if $$\displaystyle X_i $$ are independent.
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Examples
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Discrete PMF with unknown $K$:
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Solve $$\displaystyle \sum p(x) = 1 $$ for $K$.
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$$\displaystyle E(X) = \sum x p(x) $$, $$\displaystyle V(X) = E(X^2) - [E(X)]^2 $$.
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CDF: $$\displaystyle F(x) = P(X \le x) = \sum_{t \le x} p(t) $$.
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Continuous PDF: $$\displaystyle f(x)=K(1-x^2) $$ on (0,1).
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$$\displaystyle K = \left[\int_0^1 (1-x^2)dx\right]^{-1} = \frac{4}{3} $$.
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$$\displaystyle E(X) = \int_0^1 x f(x)dx $$, $$\displaystyle V(X) = \int_0^1 x^2 f(x)dx - [E(X)]^2 $$.
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4. Moments and Generating Functions
Moments
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Raw moments: $$\displaystyle \mu_r' = E(X^r) $$.
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Central moments: $$\displaystyle \mu_r = E[(X - \mu)^r] $$, where $$\displaystyle \mu = \mu_1' $$.
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Relations:
$$\displaystyle \mu_2 = \mu_2' - (\mu_1')^2 $$,
$$\displaystyle \mu_3 = \mu_3' - 3\mu_2'\mu_1' + 2(\mu_1')^3 $$,
$$\displaystyle \mu_4 = \mu_4' - 4\mu_3'\mu_1' + 6\mu_2'(\mu_1')^2 - 3(\mu_1')^4 $$.
Moment Generating Function (MGF)
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Definition: $$\displaystyle M_X(t) = E(e^{tX}) $$, exists for $t$ in some interval around 0.
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Property: $$\displaystyle M_X^{(r)}(0) = \mu_r' $$ (r-th derivative at 0).
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Use: $$\displaystyle E(X) = M_X'(0) $$, $$\displaystyle V(X) = M_X''(0) - [M_X'(0)]^2 $$.
Characteristic Function (CF)
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Definition: $$\displaystyle \varphi_X(t) = E(e^{itX}) $$, always exists.
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Relation to MGF: $$\displaystyle \varphi_X(t) = M_X(it) $$ if MGF exists.
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Normal CF: $$\displaystyle \varphi_X(t) = e^{it\mu - \frac{1}{2}\sigma^2 t^2} $$.
5. Discrete Probability Distributions
Binomial Distribution
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PMF: $$\displaystyle P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} $$, $$\displaystyle k=0,1,\dots,n $$.
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Mean: $$\displaystyle \mu = np $$, Variance: $$\displaystyle \sigma^2 = np(1-p) $$.
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MGF: $$\displaystyle M_X(t) = (1-p + pe^t)^n $$.
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Derive: $$\displaystyle M_X(t) = \sum_{k=0}^n e^{tk} \binom{n}{k} p^k (1-p)^{n-k} = (1-p + pe^t)^n $$.
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Then $$\displaystyle M_X'(t) = n(1-p+pe^t)^{n-1} \cdot pe^t $$, so $$\displaystyle M_X'(0) = np $$.
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Additivity: If $$\displaystyle X_1 \sim \text{Bin}(n_1,p) $$, $$\displaystyle X_2 \sim \text{Bin}(n_2,p) $$ independent, then $$\displaystyle X_1+X_2 \sim \text{Bin}(n_1+n_2,p) $$.
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Limiting Case → Poisson: As $n \to \infty$, $p \to 0$, $$\displaystyle np = \lambda $$ fixed:
$$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \to \frac{e^{-\lambda} \lambda^k}{k!}$$
Poisson Distribution
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PMF: $$\displaystyle P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} $$, $$\displaystyle k=0,1,2,\dots $$, $$\displaystyle \lambda > 0 $$.
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Mean & Variance: Both $$\displaystyle = \lambda $$.
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Probability calculations:
$$\displaystyle P(X=0) = e^{-\lambda} $$,
$$\displaystyle P(a \le X \le b) = \sum_{k=a}^b \frac{e^{-\lambda} \lambda^k}{k!} $$.
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Approximation to Binomial: When $n$ large, $p$ small, $$\displaystyle \lambda = np $$. Use if $n \ge 20$, $p \le 0.05$, $np \le 5$.
6. Continuous Probability Distributions
Exponential Distribution
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PDF: $$\displaystyle f(x) = \frac{1}{\theta} e^{-x/\theta} $$, $x \ge 0$ (scale $\theta$) or $$\displaystyle f(x)=\lambda e^{-\lambda x} $$ (rate $$\displaystyle \lambda=1/\theta $$).
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Mean: $\theta$, Variance: $$\displaystyle \theta^2 $$.
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MGF: $$\displaystyle M_X(t) = \frac{1}{1 - \theta t} $$ for $$\displaystyle t < 1/\theta $$.
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Memoryless Property: $$\displaystyle P(X > s+t | X > s) = P(X > t) $$.
Proof: $$\displaystyle P(X>s+t|X>s) = \frac{P(X>s+t)}{P(X>s)} = \frac{e^{-(s+t)/\theta}}{e^{-s/\theta}} = e^{-t/\theta} = P(X>t) $$.
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Special case of Gamma: $$\displaystyle \text{Gamma}(\alpha=1, \beta=\theta) $$.
Gamma Distribution
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PDF: $$\displaystyle f(x) = \frac{1}{\Gamma(\alpha)\beta^\alpha} x^{\alpha-1} e^{-x/\beta} $$, $$\displaystyle x>0 $$.
Parameters: shape $$\displaystyle \alpha > 0 $$, scale $$\displaystyle \beta > 0 $$ (or rate $$\displaystyle \lambda=1/\beta $$).
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Mean: $\alpha\beta$, Variance: $$\displaystyle \alpha\beta^2 $$.
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MGF: $$\displaystyle M_X(t) = (1 - \beta t)^{-\alpha} $$ for $$\displaystyle t < 1/\beta $$.
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Additivity: If $$\displaystyle X_i \sim \text{Gamma}(\alpha_i, \beta) $$ independent, then $$\displaystyle \sum X_i \sim \text{Gamma}(\sum \alpha_i, \beta) $$.
Normal Distribution
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PDF: $$\displaystyle f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$, $$\displaystyle -\infty < x < \infty $$.
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Properties:
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Symmetric about $\mu$, mean=median=mode.
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Inflection points at $\mu \pm \sigma$.
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$$\displaystyle P(\mu - \sigma < X < \mu + \sigma) \approx 0.68 $$, $$\displaystyle P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 0.95 $$.
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Standard Normal: $$\displaystyle Z = \frac{X - \mu}{\sigma} \sim N(0,1) $$. Use Z-tables.
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MGF: $$\displaystyle M_X(t) = e^{\mu t + \frac{1}{2}\sigma^2 t^2} $$.
Derivation: $$\displaystyle M_X(t) = E(e^{tX}) = e^{\mu t} E(e^{t(X-\mu)}) = e^{\mu t} \int_{-\infty}^{\infty} e^{ty} \frac{1}{\sqrt{2\pi}\sigma} e^{-y^2/(2\sigma^2)} dy $$, complete square in exponent.
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CF: $$\displaystyle \varphi_X(t) = e^{it\mu - \frac{1}{2}\sigma^2 t^2} $$.
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Normal Approximation to Binomial: If $n$ large, $$\displaystyle np > 5 $$, $$\displaystyle n(1-p) > 5 $$, then $X \sim \text{Bin}(n,p) \approx N(np, np(1-p))$ with continuity correction.
7. Bivariate Distributions
Joint, Marginal, Conditional
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Discrete:
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Joint PMF: $$\displaystyle p(x,y) = P(X=x, Y=y) $$.
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Marginal: $$\displaystyle p_X(x) = \sum_y p(x,y) $$.
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Conditional: $$\displaystyle p_{Y|X}(y|x) = \frac{p(x,y)}{p_X(x)} $$, $$\displaystyle p_X(x)>0 $$.
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Continuous:
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Joint PDF: $f(x,y) \ge 0$, $$\displaystyle \iint f(x,y)dxdy = 1 $$.
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Marginal: $$\displaystyle f_X(x) = \int_{-\infty}^{\infty} f(x,y)dy $$.
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Conditional: $$\displaystyle f_{Y|X}(y|x) = \frac{f(x,y)}{f_X(x)} $$, $$\displaystyle f_X(x)>0 $$.
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Independence
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Definition: $X,Y$ independent iff $$\displaystyle p(x,y)=p_X(x)p_Y(y) $$ (discrete) or $$\displaystyle f(x,y)=f_X(x)f_Y(y) $$ (continuous) for all $x,y$.
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Implication: If independent, $$\displaystyle E(XY) = E(X)E(Y) $$.
Bivariate Normal Distribution
- Joint PDF:
$$f(x,y) = \frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}} \exp\left(-\frac{1}{2(1-\rho^2)}\left[\frac{(x-\mu_1)^2}{\sigma_1^2} - 2\rho\frac{(x-\mu_1)(y-\mu_2)}{\sigma_1\sigma_2} + \frac{(y-\mu_2)^2}{\sigma_2^2}\right]\right)$$
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Marginals: $$\displaystyle X \sim N(\mu_1, \sigma_1^2) $$, $$\displaystyle Y \sim N(\mu_2, \sigma_2^2) $$.
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Conditional: $$\displaystyle X|Y=y \sim N\left(\mu_1 + \rho\frac{\sigma_1}{\sigma_2}(y-\mu_2), \sigma_1^2(1-\rho^2)\right) $$.
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Independence iff $$\displaystyle \rho = 0 $$.
8. Correlation and Regression Analysis
Correlation
- Karl Pearson’s $r$:
$$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} = \frac{\text{Cov}(X,Y)}{s_x s_y}$$
Properties: $-1 \le r \le 1$, unitless, symmetric $$\displaystyle r_{xy}=r_{yx} $$.
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Spearman’s Rank Correlation $\rho$:
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Assign ranks to $X$ and $Y$ separately (average ranks for ties).
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$$\displaystyle \rho = 1 - \frac{6\sum d_i^2}{n(n^2-1)} $$, where $$\displaystyle d_i = \text{rank}(x_i) - \text{rank}(y_i) $$.
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Maximum value = 1 (achieved when ranks perfectly monotonic).
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Regression
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Simple Linear Regression: $$\displaystyle y = a + bx $$.
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Least Squares Estimates:
$$b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}, \quad a = \bar{y} - b\bar{x}$$
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Properties:
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Line passes through $(\bar{x}, \bar{y})$.
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Minimizes $$\displaystyle \sum (y_i - \hat{y}_i)^2 $$.
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Normal Equations (for $$\displaystyle y = a + bx $$):
$$\displaystyle \sum y = na + b\sum x $$,
$$\displaystyle \sum xy = a\sum x + b\sum x^2 $$.
Fitting Parabola ($$\displaystyle y = a + bx + cx^2 $$)
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Normal Equations:
$$\displaystyle \sum y = na + b\sum x + c\sum x^2 $$,
$$\displaystyle \sum xy = a\sum x + b\sum x^2 + c\sum x^3 $$,
$$\displaystyle \sum x^2 y = a\sum x^2 + b\sum x^3 + c\sum x^4 $$.
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Solve simultaneously for $a,b,c$.
Correlation vs. Regression
| Feature | Correlation | Regression |
|---|---|---|
| Purpose | Measure strength/direction of linear association | Predict $Y$ from $X$ |
| Symmetry | $$\displaystyle r_{xy}=r_{yx} $$ | Not symmetric: $y$ on $x$ vs. $x$ on $y$ |
| Units | Unitless | Dependent on units of $X,Y$ |
9. Statistical Inference
Sampling & Estimation
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Central Limit Theorem (CLT): For large $n$, $$\displaystyle \bar{X} \sim N(\mu, \sigma^2/n) $$ regardless of population distribution.
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Confidence Interval for $\mu$ (large samples, $\sigma$ known):
$$\displaystyle \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} $$.
If $\sigma$ unknown, use $s$ and $t$-distribution (small $n$ normal population).
Hypothesis Testing
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z-test for Single Mean (large $n$, $\sigma$ known):
$$\displaystyle z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} $$, compare with $$\displaystyle z_{\alpha} $$.
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z-test for Difference of Means (large independent samples, $$\displaystyle \sigma_1,\sigma_2 $$ known):
$$\displaystyle z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} $$.
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Test for Single Proportion (normal approx to binomial):
$$\displaystyle z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}} $$, where $$\displaystyle \hat{p}=x/n $$.
Chi-square ($$\displaystyle \chi^2 $$) Test
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Chi-square Distribution:
PDF: $$\displaystyle f(x) = \frac{1}{2^{\nu/2}\Gamma(\nu/2)} x^{\nu/2 - 1} e^{-x/2} $$, $$\displaystyle x>0 $$, $\nu$ = degrees of freedom.
Mean $$\displaystyle = \nu $$, Variance $$\displaystyle = 2\nu $$.
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Goodness of Fit Test:
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$$\displaystyle H_0 $$: Observed frequencies follow a specified distribution.
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Test statistic: $$\displaystyle \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} $$.
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df = (number of categories $-$ 1 $-$ number of parameters estimated from data).
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Conditions: (i) Expected frequencies $$\displaystyle E_i \ge 5 $$ (combine if needed), (ii) independent observations, (iii) categorical data.
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Test for Independence (contingency table):
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$$\displaystyle H_0 $$: Two categorical variables are independent.
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$$\displaystyle \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} $$, where $$\displaystyle E_{ij} = \frac{(\text{row total})_i \times (\text{column total})_j}{\text{grand total}} $$.
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df = $(r-1)(c-1)$.
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Chebyshev’s Inequality
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Statement: For any $$\displaystyle k>0 $$, $$\displaystyle P(|X - \mu| \ge k\sigma) \le \frac{1}{k^2} $$.
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Proof Outline: Apply Markov’s inequality to $$\displaystyle Y = (X-\mu)^2 $$:
$$\displaystyle P(|X-\mu| \ge k\sigma) = P((X-\mu)^2 \ge k^2\sigma^2) \le \frac{E[(X-\mu)^2]}{k^2\sigma^2} = \frac{\sigma^2}{k^2\sigma^2} = \frac{1}{k^2} $$.
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Application: Distribution-free bound (works for any distribution with finite $\mu,\sigma$).
10. Special Topics and Applications
Approximations
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Poisson Approximation to Binomial: $n \to \infty$, $p \to 0$, $$\displaystyle np = \lambda $$ fixed.
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Normal Approximation to Binomial: $$\displaystyle np > 5 $$, $$\displaystyle n(1-p) > 5 $$. Use continuity correction: $$\displaystyle P(X \le k) \approx P\left(Z \le \frac{k+0.5 - np}{\sqrt{np(1-p)}}\right) $$.
Solving for Normal Parameters from Percentiles
Given $$\displaystyle P(X < x_1) = p_1 $$, $$\displaystyle P(X < x_2) = p_2 $$:
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Find $$\displaystyle z_1, z_2 $$ from standard normal table such that $$\displaystyle \Phi(z_1)=p_1 $$, $$\displaystyle \Phi(z_2)=p_2 $$.
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Solve: $$\displaystyle z_1 = \frac{x_1 - \mu}{\sigma} $$, $$\displaystyle z_2 = \frac{x_2 - \mu}{\sigma} $$ simultaneously for $\mu, \sigma$.
Probability Calculations
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Binomial: $$\displaystyle P(X \ge k) = 1 - P(X \le k-1) = 1 - \sum_{i=0}^{k-1} \binom{n}{i} p^i (1-p)^{n-i} $$.
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Poisson: Cumulative from tables or software.
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Normal: Standardize $$\displaystyle Z = (X-\mu)/\sigma $$, use Z-table.
Type I & Type II Errors (brief)
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Type I: Reject $$\displaystyle H_0 $$ when true (probability = $\alpha$).
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Type II: Accept $$\displaystyle H_0 $$ when false (probability = $\beta$).
11. Additional Topics from Past Papers
Partial and Multiple Correlation Coefficients
- Partial Correlation: Correlation between two variables after removing linear effect of others. For three variables $X,Y,Z$:
$$r_{XY.Z} = \frac{r_{XY} - r_{XZ}r_{YZ}}{\sqrt{(1-r_{XZ}^2)(1-r_{YZ}^2)}}$$
- Multiple Correlation: Correlation between one variable and its best linear predictor from several others. $$\displaystyle R_{Y.X_1,X_2} = \sqrt{1 - \frac{\text{Var}(Y|\text{regression})}{\text{Var}(Y)}} $$.
Bayes’ Theorem with Multiple Partitions
Example: Defective items from factories A, B, C with productions $$\displaystyle N_A, N_B, N_C $$ and defect rates $$\displaystyle p_A, p_B, p_C $$.
$$P(\text{from A}|\text{defective}) = \frac{N_A p_A}{N_A p_A + N_B p_B + N_C p_C}$$
Hypergeometric Expectation
Population: $N$ items, $K$ defectives. Sample $n$ without replacement. $$\displaystyle X = $$ number of defectives in sample $\sim \text{Hypergeometric}(N,K,n)$. $$\displaystyle E(X) = n \cdot \frac{K}{N} $$ (same as binomial mean, but without replacement).
F-test for Equality of Variances
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Test $$\displaystyle H_0: \sigma_1^2 = \sigma_2^2 $$ vs. $$\displaystyle H_1: \sigma_1^2 \ne \sigma_2^2 $$.
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Test statistic: $$\displaystyle F = \frac{s_1^2}{s_2^2} $$ (larger variance in numerator).
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Under $$\displaystyle H_0 $$, $$\displaystyle F \sim F(n_1-1, n_2-1) $$.
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Compare with critical $$\displaystyle F_{\alpha/2} $$ or $$\displaystyle F_{1-\alpha/2} $$ (two-tailed).
[!TIP] In two-sample z-tests for means, if $$\displaystyle \sigma_1, \sigma_2 $$ unknown but large samples, use $$\displaystyle s_1, s_2 $$ as approximations. For small samples from normal populations, use t-test with pooled variance if variances equal.