Statistics & Probability: notes and practice questions

Qualitative data is descriptive, non-numerical. Samples may not be representative of the entire population; minimise bias with random sampling.

Univariate Data: Data in a single variable. Mode: The most common value.

Measures of Central Tendency:. Mode: Most frequent value.

Bivariate Data: Data on two variables, paired to examine relationships. Correlation vs. Causation: Correlation does not imply causation.

**Expected Value ($E(X)$):** The mean of a random variable $X$. Expected Number of Occurrences: For $n$ trials with success probability $p$, expected occurrences are $np$.

Probability uses diagrams and notation to model combined events. Mutually Exclusive Events: Cannot occur together. $P(A \cap B) = 0$. So, $P(A \cup B) = P(A) + P(B)$.

Conditional Probability: Event A occurs given event B has already occurred. Independent Events: Occurrence of one does not affect the probability of the other; $P(A|B) = P(A)$ and $P(B|A) = P(B)$.

A random variable's value depends on a random event's outcome. A discrete random variable takes specific, separate values (e.g., non-negative integers).

A discrete random variable's value depends on a random event, unknown until carried out. A discrete probability distribution lists all possible discrete outcomes and their probabilities.

Normal distribution is a continuous probability distribution. Notation: $X \sim N(\mu, \sigma^2)$, where $\mu$ is the population mean and $\sigma^2$ is the population variance.

Spearman's Rank Correlation Coefficient ($r_s$) measures the strength and direction of a monotonic relationship between two variables. A monotonic function either only increases or only decreases.

Hypothesis Test: Uses sample data to test a statement about a population. Null Hypothesis ($H_0$): Assumes no difference, no change, no association, or data follows a specific distribution.

Qualitative Data: Expressed in words, grouped into categories for statistical analysis. Chi-squared Test for Independence: Determines association between two categorical variables; null hypothesis: variables are independent.

Non-linear regression: Used when a curve fits bivariate data better than a straight line. Residual: Difference between actual and predicted value: $\text{Residual}_i = y_i - f(x_i)$.

An estimator is a random variable; an estimate is its numerical value from a sample. An estimator is unbiased (HL) if its expected value equals the population parameter: $E(\text{Estimator}) = \text{Parameter}$.

For a single random variable `X` and constants `a, b`: $E(aX + b) = aE(X) + b$.

Confidence Intervals for the Mean (with Sample Mean Distribution & Central Limit Theorem) are HL syllabus. The sample mean ($\bar{x}$) is used as a point estimate for the population mean ($\mu$).

Poisson Distribution: Discrete probability distribution modeling the number of occurrences of an event in a fixed interval. Conditions: Occurrences are independent and happen at a uniform average rate.

Hypothesis Test: Uses sample data to evaluate a statement about a population parameter. Null Hypothesis ($H_0$): Default statement (no change, no difference, or no correlation).

State: A mutually exclusive event that can change over time. Markov Chain: A mathematical model describing a sequence of states over discrete time steps.