Probability Definition:

The probability of an event A, denoted \( P(A) \), is a number between 0 and 1 that represents the likelihood of event A occurring.

\[ 0 \leq P(A) \leq 1 \]

Key Terms:

• Random Experiment: A process with uncertain outcomes
• Sample Space (S): The set of all possible outcomes
• Event: A subset of the sample space
• Probability: The likelihood of an event occurring

Classical Probability:

\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

1. Complement Rule:

\[ P(A^c) = 1 - P(A) \]

Where \( A^c \) is the complement of event A (A does not occur)

2. Addition Rule (Mutually Exclusive Events):

For events that cannot occur simultaneously:

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

3. General Addition Rule:

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

4. Multiplication Rule (Independent Events):

For events where one doesn't affect the other:

\[ P(A \cap B) = P(A) \times P(B) \]

5. Conditional Probability:

The probability of A given that B has occurred:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Bayes' Theorem: A fundamental formula for updating probabilities based on new information

\[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \]

Extended Form:

\[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B|A) \times P(A) + P(B|A^c) \times P(A^c)} \]

Applications: Medical diagnosis, spam filtering, machine learning, decision analysis

Expected Value (Mean) - Discrete:

\[ E(X) = \mu = \sum_{i} x_i P(X = x_i) \]

Expected Value - Continuous:

\[ E(X) = \mu = \int_{-\infty}^{\infty} x f(x) \, dx \]

Variance - Discrete:

\[ \text{Var}(X) = \sigma^2 = \sum_{i} (x_i - \mu)^2 P(X = x_i) \]

Alternative Formula:

\[ \text{Var}(X) = E(X^2) - [E(X)]^2 \]

Standard Deviation:

\[ \sigma = \sqrt{\text{Var}(X)} \]

Models a single trial with two outcomes: success (1) or failure (0)

Parameters: \( p \) = probability of success

PMF:

\[ P(X = x) = p^x(1-p)^{1-x}, \quad x \in \{0, 1\} \]

Mean: \( E(X) = p \)

Variance: \( \text{Var}(X) = p(1-p) \)

Models the number of successes in \( n \) independent Bernoulli trials

Notation: \( X \sim \text{Binomial}(n, p) \)

PMF:

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

Where \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Mean: \( E(X) = np \)

Variance: \( \text{Var}(X) = np(1-p) \)

Models the number of events occurring in a fixed interval of time or space

Notation: \( X \sim \text{Poisson}(\lambda) \)

PMF:

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

Mean: \( E(X) = \lambda \)

Variance: \( \text{Var}(X) = \lambda \)

Applications: Modeling rare events, call centers, radioactive decay

All values in an interval are equally likely

Notation: \( X \sim \text{Uniform}(a, b) \)

PDF:

\[ f(x) = \frac{1}{b-a}, \quad a \leq x \leq b \]

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

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

The most important continuous distribution, characterized by its bell-shaped curve

Notation: \( X \sim N(\mu, \sigma^2) \)

PDF:

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

Mean: \( E(X) = \mu \)

Variance: \( \text{Var}(X) = \sigma^2 \)

Standard Normal Distribution: \( Z \sim N(0, 1) \)

Standardization:

\[ Z = \frac{X - \mu}{\sigma} \]

Empirical Rule (68-95-99.7 Rule):

• 68% of data falls within 1 standard deviation of the mean
• 95% of data falls within 2 standard deviations of the mean
• 99.7% of data falls within 3 standard deviations of the mean

Models the time between events in a Poisson process

Notation: \( X \sim \text{Exp}(\lambda) \)

PDF:

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

Mean: \( E(X) = \frac{1}{\lambda} \)

Variance: \( \text{Var}(X) = \frac{1}{\lambda^2} \)

1. Range:

\[ \text{Range} = \text{Maximum} - \text{Minimum} \]

2. Variance (Sample):

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

3. Standard Deviation (Sample):

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

4. Interquartile Range (IQR):

\[ \text{IQR} = Q_3 - Q_1 \]

Where \( Q_1 \) is the 25th percentile and \( Q_3 \) is the 75th percentile

Central Limit Theorem (CLT):

For a large sample size \( n \), the sampling distribution of the sample mean \( \bar{X} \) is approximately normal, regardless of the population's distribution:

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

Standard Error of the Mean:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

Key Points:

• Generally, \( n \geq 30 \) is considered large enough
• The CLT is fundamental to statistical inference
• Allows us to use normal distribution methods even when the population is not normal

A confidence interval provides a range of plausible values for a population parameter

Confidence Interval for Population Mean (\( \sigma \) known):

\[ \bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} \]

Confidence Interval for Population Mean (\( \sigma \) unknown):

\[ \bar{x} \pm t^* \frac{s}{\sqrt{n}} \]

Where \( t^* \) is from the t-distribution with \( n-1 \) degrees of freedom

Common Confidence Levels:

• 90% confidence: \( z^* = 1.645 \)
• 95% confidence: \( z^* = 1.96 \)
• 99% confidence: \( z^* = 2.576 \)

Interpretation: We are 95% confident that the true population parameter lies within the calculated interval.

When to use: Testing a claim about a population mean when \( \sigma \) is known and \( n \) is large

Test Statistic:

\[ z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} \]

Hypotheses:

• Two-tailed: \( H_0: \mu = \mu_0 \) vs \( H_a: \mu \neq \mu_0 \)
• Right-tailed: \( H_0: \mu = \mu_0 \) vs \( H_a: \mu > \mu_0 \)
• Left-tailed: \( H_0: \mu = \mu_0 \) vs \( H_a: \mu < \mu_0 \)

When to use: Testing a claim about a population mean when \( \sigma \) is unknown

Test Statistic:

\[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \]

Follows a t-distribution with \( n-1 \) degrees of freedom

When to use: Comparing means of two independent groups

Test Statistic:

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

When to use: Comparing two related samples (before/after, matched pairs)

Test Statistic:

\[ t = \frac{\bar{d} - 0}{s_d/\sqrt{n}} \]

Where \( \bar{d} \) is the mean of differences and \( s_d \) is the standard deviation of differences

Pearson Correlation Coefficient (r):

Measures the strength and direction of linear relationship between two variables

\[ r = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2} \sqrt{\sum_{i=1}^n (y_i - \bar{y})^2}} \]

Properties:

• \( -1 \leq r \leq 1 \)
• \( r = 1 \): Perfect positive linear relationship
• \( r = -1 \): Perfect negative linear relationship
• \( r = 0 \): No linear relationship

Coefficient of Determination:

\[ r^2 = \text{proportion of variance explained} \]

Regression Equation:

\[ \hat{y} = a + bx \]

Slope (b):

\[ b = r \frac{s_y}{s_x} = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sum(x_i - \bar{x})^2} \]

Intercept (a):

\[ a = \bar{y} - b\bar{x} \]

Interpretation:

• \( b \): Change in y for a one-unit change in x
• \( a \): Predicted value of y when x = 0