📊AP Statistics Unit 4 – Probability, Random Variables, and Probability Distributions

Key Concepts and Definitions

• Probability quantifies the likelihood of an event occurring and ranges from 0 (impossible) to 1 (certain)
• Sample space ($S$) consists of all possible outcomes of an experiment or random process
• An event ($E$) is a subset of the sample space containing one or more outcomes
  • Complement of an event ($E'$ or $E^c$) contains all outcomes in the sample space that are not in the event
• Random variables ($X$) are functions that assign a numerical value to each outcome in a sample space
  • Discrete random variables have countable distinct values (integers)
  • Continuous random variables can take on any value within an interval
• Probability distributions describe how probabilities are distributed over the values of a random variable
  • Probability mass functions (PMF) are used for discrete random variables
  • Probability density functions (PDF) are used for continuous random variables

Types of Probability

• Classical probability is based on the assumption that all outcomes in the sample space are equally likely
  • Calculated as $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$
• Empirical probability relies on observed data and is calculated as the relative frequency of an event
  • Calculated as $P(E) = \frac{\text{number of times event E occurs}}{\text{total number of trials}}$
• Subjective probability is based on an individual's belief or judgment about the likelihood of an event
• Conditional probability is the probability of an event occurring given that another event has already occurred
  • Denoted as $P(A|B)$ and read as "the probability of A given B"
  • Calculated as $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $P(B) \neq 0$
• Joint probability is the probability of two or more events occurring simultaneously
  • For independent events, $P(A \cap B) = P(A) \times P(B)$
  • For dependent events, $P(A \cap B) = P(A) \times P(B|A)$

Random Variables Explained

• Random variables are used to quantify the outcomes of a random experiment
• The probability distribution of a random variable describes the probabilities associated with each possible value
• Expected value (mean) of a random variable is the average value obtained if the experiment is repeated many times
  • For a discrete random variable, $E(X) = \sum_{x} x \cdot P(X=x)$
  • For a continuous random variable, $E(X) = \int_{-\infty}^{\infty} x \cdot f(x) dx$
• Variance measures the spread of a random variable around its expected value
  • $Var(X) = E[(X - \mu)^2]$, where $\mu = E(X)$
• Standard deviation is the square root of the variance and has the same units as the random variable
  • $\sigma = \sqrt{Var(X)}$

Probability Distributions Overview

• Probability distributions assign probabilities to the possible values of a random variable
• Discrete probability distributions are used for random variables with countable outcomes (coin flips, dice rolls)
• Continuous probability distributions are used for random variables with an infinite number of possible values within an interval (heights, weights)
• Cumulative distribution functions (CDF) give the probability that a random variable is less than or equal to a specific value
  • For discrete random variables, $F(x) = P(X \leq x) = \sum_{t \leq x} P(X = t)$
  • For continuous random variables, $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$
• The probability density function (PDF) for a continuous random variable is the derivative of its CDF
  • $f(x) = F'(x)$

Common Probability Distributions

• Bernoulli distribution models a single trial with two possible outcomes (success or failure)
  • $P(X = 1) = p$ and $P(X = 0) = 1 - p$, where $p$ is the probability of success
• Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials
  • $X \sim B(n, p)$, where $n$ is the number of trials and $p$ is the probability of success
  • $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
• Poisson distribution models the number of rare events occurring in a fixed interval of time or space
  • $X \sim Poisson(\lambda)$, where $\lambda$ is the average rate of occurrence
  • $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$
• Normal (Gaussian) distribution is a continuous distribution with a bell-shaped curve
  • $X \sim N(\mu, \sigma^2)$, where $\mu$ is the mean and $\sigma^2$ is the variance
  • PDF: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
• Standard normal distribution is a normal distribution with $\mu = 0$ and $\sigma = 1$
  • $Z = \frac{X - \mu}{\sigma}$ is used to standardize any normal random variable $X$

Calculating Probabilities

• For discrete random variables, probabilities are calculated using the probability mass function (PMF)
  • $P(X = x)$ is the probability that the random variable $X$ takes on the specific value $x$
• For continuous random variables, probabilities are calculated using the probability density function (PDF) and integration
  • $P(a \leq X \leq b) = \int_{a}^{b} f(x) dx$
• Complement rule: $P(E') = 1 - P(E)$
• Addition rule for mutually exclusive events: $P(A \cup B) = P(A) + P(B)$
• Multiplication rule for independent events: $P(A \cap B) = P(A) \times P(B)$
• Bayes' theorem is used to calculate conditional probabilities
  • $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$, where $P(B) \neq 0$

Real-World Applications

• Quality control uses probability distributions to model the number of defective items in a production process (binomial, Poisson)
• Insurance companies use probability distributions to model claim amounts and frequencies (normal, exponential)
• Financial markets use probability distributions to model stock prices and returns (normal, lognormal)
• Polling and surveys use probability distributions to model the proportion of people with a certain opinion or characteristic (binomial)
• Queuing theory uses probability distributions to model waiting times and queue lengths (exponential, Poisson)

Practice Problems and Tips

• Identify the type of probability distribution based on the given information and context
• Determine the parameters of the distribution (e.g., $n$ and $p$ for binomial, $\lambda$ for Poisson, $\mu$ and $\sigma$ for normal)
• Use the appropriate formula or table to calculate probabilities or find values of the random variable
  • For the normal distribution, use the standard normal table with $z$-scores
• Be cautious when working with continuous random variables, as probabilities are calculated using integration and areas under the curve
• Practice solving problems using various probability rules, such as the complement rule, addition rule, and multiplication rule
• Understand the assumptions behind each probability distribution and check if they are appropriate for the given problem
• When solving conditional probability problems, clearly identify the given information and the event of interest
• Double-check your calculations and ensure that your final answer makes sense in the context of the problem