Statistics & Probability: notes and practice questions

Population: The entire set of data or individuals being studied. Sampling Techniques:

Frequency distribution tables organize data into intervals (bins) with corresponding frequencies. Histograms represent data as bars to show frequency distribution.

Central Tendency: Mean: Average of the data.

Scatter diagrams display pairs of data points to show the relationship between two variables. The line of best fit (or regression line) represents the linear relationship, typically found using the least squares method.

Probability of an event (P(E)): $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}}$.

Venn diagrams represent sets and probabilities visually, showing intersections ($A \cap B$), unions ($A \cup B$), and complements ($A'$). Tree diagrams display sequential events and their probabilities.

Conditional Probability: The probability of event $A$ given that $B$ has occurred: $P(A|B) = \frac{P(A \cap B)}{P(B)}, \, P(B) \neq 0$.

A discrete random variable takes specific, countable values, each with a probability $P(X=x)$. The sum of probabilities is always 1:

Describes the probability of $x$ successes in $n$ independent trials of a binary event (success/failure) with probability of success $p$. Probability mass function:

A continuous, symmetric, bell-shaped curve representing data distributed around a mean ($\mu$) with standard deviation ($\sigma$). Probability is calculated using the z-score:

Regression line (y on x): Predicts $y$ using $x$, expressed as $y = a + bx$. Reverse regression (x on y): Predicts $x$ using $y$, expressed as $x = c + dy$.

Standardising converts a normal variable $X$ with mean $\mu$ and standard deviation $\sigma$ to a standard normal variable $Z$: $Z = \frac{X - \mu}{\sigma}$.

Binomial Distribution: Models the number of successes in a fixed number of independent trials with constant success probability ($n$, $p$). Key stats: Mean = $np$, Variance = $np(1-p)$. Use Binomial PDF for exact probabilities and CDF for cumulative ones. Normal Distribution: A continuous distribution, denoted $X \backsim N(\mu, \sigma^2)$. Characterized by its bell curve. Standardize using $Z = (X - \mu) / \sigma$ to get $Z \backsim N(0, 1)$. Use Normal CDF to find probabilities (areas) and Inverse Normal to find values given probabilities. Empirical rule: ~68% within $\mu \pm \sigma$, ~95% within $\mu \pm 2\sigma$, ~99.7% within $\mu \pm 3\sigma$.

Random variables can be discrete (countable) or continuous (measurable). For discrete variables, use Probability Mass Functions (PMF) to define probabilities, and calculate mean and variance using summations.