Statistics Exam Study Guide

Probability Exam Topics Breakdown

1. Random Events and Probability Properties

• Random Event: A subset of sample space (e.g., rolling an even number).
• Elementary Event: A single outcome (e.g., rolling a 4).
• Sure Event: An event that always occurs (entire sample space ( \Omega )).
• Impossible Event: An event that never occurs (empty set ( \emptyset )).
• Properties:
  • ( P(\Omega) = 1 )
  • ( P(\emptyset) = 0 )
  • ( P(A^c) = 1 - P(A) )
  • ( P(A \cup B) = P(A) + P(B) - P(A \cap B) )
• Classical Probability: ( P(A) = \frac{|A|}{|\Omega|} )
• Statistical Probability: ( P(A) \approx \frac{n_A}{n} )

2. Conditional Probability and Independence

• Conditional Probability: ( P(A|B) = \frac{P(A \cap B)}{P(B)} )
• Independence: ( P(A \cap B) = P(A)P(B) )
• Property: ( P(A^c|B) = 1 - P(A|B) )

3. Law of Total Probability & Bayes Theorem

• Total Probability: ( P(B) = \sum P(A_i)P(B|A_i) )
• Bayes Theorem: ( P(A_i|B) = \frac{P(B|A_i)P(A_i)}{\sum P(B|A_j)P(A_j)} )

4. Random Variables, CDF, Mean/Variance of Transformations

• CDF: ( F(x) = P(X \le x) )
• Transformations:
  • ( \mathbb{E}[aX + b] = a\mathbb{E}[X] + b )
  • ( \text{Var}(aX + b) = a^2 \text{Var}(X) )

5. Discrete Random Variables, PMF, Mean and Variance

• PMF: ( p(x) = P(X = x) ), ( \sum p(x) = 1 )
• Mean: ( \mathbb{E}[X] = \sum x p(x) )
• Variance: ( \text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 )

6. Continuous Random Variables, PDF, Mean, Variance, Quantiles

• PDF: ( f(x) \ge 0 ), ( \int f(x) dx = 1 )
• Mean: ( \mathbb{E}[X] = \int x f(x) dx )
• Variance: ( \text{Var}(X) = \int x^2 f(x) dx - (\mathbb{E}[X])^2 )
• Quantile: ( u_\alpha ) such that ( P(X \le u_\alpha) = \alpha )

7. Covariance, Correlation, Random Vectors

• Covariance: ( \text{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y] )
• Correlation: ( \rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} )
• Variance of Sum: ( \text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y) )

8. Asymptotic Behavior, LLN, CLT

• SLLN: Sample mean ( \bar{X}_n \to \mu ) almost surely.
• CLT: ( Z_n = \frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \to N(0, 1) )

9. Normal & Standard Normal Distribution

• ( X \sim N(\mu, \sigma^2) )
• Standardized: ( Z = \frac{X - \mu}{\sigma} )

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Statistics Exam Topics Breakdown

1. Random Sample, Normal Distribution, Sample Mean/Variance

• Sample Mean: ( \bar{X} = \frac{1}{n} \sum X_i )
• Sample Variance: ( S^2 = \frac{1}{n-1} \sum (X_i - \bar{X})^2 )
• Normal Distribution of Sample Mean: ( \bar{X} \sim N(\mu, \sigma^2/n) )
• Chi-square Distribution: ( \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1) )

2. Point/Interval Estimators, Unbiased and Best Estimators

• Unbiased Estimator: ( \mathbb{E}[\hat{\theta}] = \theta )
• Confidence Interval (known ( \sigma )): ( \bar{X} \pm z \cdot \frac{\sigma}{\sqrt{n}} )
• Confidence Interval (unknown ( \sigma )): ( \bar{X} \pm t \cdot \frac{S}{\sqrt{n}} )

3. Hypothesis Testing Basics

• Null/Alternative Hypotheses: ( H_0, H_1 )
• Type I Error ( \alpha ), Type II Error ( \beta )
• p-value: Probability of observing extreme result under ( H_0 )

4. One-Sample Tests (Mean & Variance)

• Mean Test: ( T = \frac{\bar{X} - \mu_0}{S / \sqrt{n}} \sim t(n-1) )
• Variance Test: ( T = \frac{(n-1)S^2}{\sigma_0^2} \sim \chi^2(n-1) )
• CI Rejection: If ( \mu_0 \notin \text{CI} ), reject ( H_0 )

5. Two-sample t-test

• Equal Variances Assumed: ( T = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{S_p^2(\frac{1}{n_1} + \frac{1}{n_2})}} )
• Pooled Variance: ( S_p^2 = \frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2} )

6. Two-sample Test of Variance

• F-test: ( F = \frac{S_1^2}{S_2^2} \sim F(n_1 - 1, n_2 - 1) )

7. Paired Difference Test

• Differences: ( D_i = X_i - Y_i )
• t-test: ( T = \frac{\bar{D}}{S_D / \sqrt{n}} )

8. ANOVA

• Used to compare 3+ group means.
• Assumptions: Normality, Equal variances, Independence.

9. Correlation Test

• Test Statistic: ( T = \frac{r \sqrt{n-2}}{\sqrt{1 - r^2}} \sim t(n-2) )
• Chi-square test (( \chi^2 )): Tests if observed matches expected.

12. Linear Regression

• Model: ( Y = \beta_0 + \beta_1 X + \varepsilon ), ( \varepsilon \sim N(0, \sigma^2) )
• Estimators: ( \hat{\beta}_1, \hat{\beta}_0 )
• ( R^2 ): Proportion of variance in ( Y ) explained by ( X ).
• Confidence bands for predictions widen away from ( \bar{x} ).