📌 Descriptive Statistics

Mean (Average):

\( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \)

Median:

Middle value of ordered data

Mode:

Most frequent value

Variance (Sample):

\( s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \)

Standard Deviation (Sample):

\( s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \)

Population Variance:

\( \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \)

Population Standard Deviation:

\( \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \)

Interquartile Range (IQR):

\( IQR = Q_3 - Q_1 \)

📌 Probability Basics

Probability of an Event:

\( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \)

Complement Rule:

\( P(A') = 1 - P(A) \)

Addition Rule (Mutually Exclusive):

\( P(A \cup B) = P(A) + P(B) \)

Addition Rule (General):

\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

Multiplication Rule (Independent):

\( P(A \cap B) = P(A) \cdot P(B) \)

Conditional Probability:

\( P(A|B) = \frac{P(A \cap B)}{P(B)} \), if \( P(B) > 0 \)

📌 Discrete Distributions

Binomial Probability:

\( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)

Binomial Mean:

\( E(X) = np \)

Binomial Variance:

\( Var(X) = np(1-p) \)

Poisson Probability:

\( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \)

Poisson Mean and Variance:

\( E(X) = \lambda, \, Var(X) = \lambda \)

📌 Continuous Distributions

Uniform Distribution PDF:

\( f(x) = \frac{1}{b-a} \), for \( a \leq x \leq b \)

Uniform Mean:

\( E(X) = \frac{a + b}{2} \)

Uniform Variance:

\( Var(X) = \frac{(b-a)^2}{12} \)

Normal Distribution PDF:

\( f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)

Normal Mean and Variance:

\( E(X) = \mu, \, Var(X) = \sigma^2 \)

Exponential PDF:

\( f(x) = \lambda e^{-\lambda x} \), \( x \geq 0 \)

Exponential Mean and Variance:

\( E(X) = \frac{1}{\lambda}, \, Var(X) = \frac{1}{\lambda^2} \)

📌 Central Limit Theorem

Sample Mean Distribution:

\( \bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \), for large \( n \)

Standard Error of the Mean:

\( SE = \frac{\sigma}{\sqrt{n}} \)

Z-Score for Sample Mean:

\( Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \)

📌 Confidence Intervals

Confidence Interval for Mean (Known Variance):

\( \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}} \)

Confidence Interval for Mean (Unknown Variance):

\( \bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}} \)

Confidence Interval for Proportion:

\( \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)

📌 Hypothesis Testing

Z-Test Statistic (Mean):

\( Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \)

t-Test Statistic (Mean):

\( t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \)

Z-Test Statistic (Proportion):

\( Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \)

P-Value (General):

Probability of observing test statistic as extreme as calculated

📌 Regression Analysis

Simple Linear Regression:

\( y = \beta_0 + \beta_1 x + \epsilon \)

Slope (\( \beta_1 \)):

\( \beta_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \)

Intercept (\( \beta_0 \)):

\( \beta_0 = \bar{y} - \beta_1 \bar{x} \)

Coefficient of Determination (\( R^2 \)):

\( R^2 = 1 - \frac{\sum (y_i - \hat{y}_i)^2}{\sum (y_i - \bar{y})^2} \)

📌 ANOVA

Total Sum of Squares (SST):

\( SST = \sum (y_i - \bar{y})^2 \)

Between-Group Sum of Squares (SSB):

\( SSB = \sum n_k (\bar{y}_k - \bar{y})^2 \)

Within-Group Sum of Squares (SSW):

\( SSW = \sum \sum (y_{ik} - \bar{y}_k)^2 \)

F-Statistic:

\( F = \frac{MSB}{MSW} = \frac{SSB / (k-1)}{SSW / (N-k)} \)

📌 Bayesian Statistics

Bayes’ Theorem:

\( P(A|B) = \frac{P(B|A) P(A)}{P(B)} \)

Posterior Distribution:

\( P(\theta|D) = \frac{P(D|\theta) P(\theta)}{\int P(D|\theta) P(\theta) \, d\theta} \)

Expected Value (Continuous):

\( E(\theta) = \int \theta P(\theta|D) \, d\theta \)