Statistics Practice Test Cheat Sheet

This cheat sheet covers fundamental concepts in probability and statistics, including disjoint and independent events, probability distributions, binomial and Poisson distributions, and expected value calculations.

Core Principles

• Disjoint events cannot occur simultaneously (P(A and B) = 0).
• Independent events do not affect each other's probability (P(A|B) = P(A)).
• Probability distributions require non-negative probabilities that sum to 1.
• Binomial distribution applies to a fixed number of independent trials with two outcomes.
• Poisson distribution models the number of events in a fixed interval with a known average rate.

Action Steps

• Identify if events are disjoint or independent.
• Calculate probabilities using appropriate formulas (union, intersection, conditional).
• Verify if a distribution is a valid probability distribution.
• Determine if a scenario fits a binomial or Poisson distribution.
• Calculate mean and standard deviation for distributions.
• Use cumulative probabilities (CDF) for 'at least' or 'less than' scenarios.
• Calculate expected value to determine average outcomes.

Formulas

• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• $P(A|B) = \frac{P(A \cap B)}{P(B)}$
• Mean of Binomial Distribution: $\mu = np$
• Standard Deviation of Binomial Distribution: $\sigma = \sqrt{npq}$
• Expected Value: $E(X) = \sum [x \cdot P(x)]$
• Poisson Probability: $P(x; \lambda) = \frac{\lambda^x e^{-\lambda}}{x!}$

Key Terms

• Disjoint Events: Events that cannot occur at the same time.
• Independent Events: The occurrence of one event does not affect the probability of another.
• Probability Distribution: A function that describes the likelihood of obtaining the possible values of a random variable.
• Binomial Distribution: Probability distribution for a fixed number of independent trials with two outcomes.
• Poisson Distribution: Probability distribution for the number of events in a fixed interval of time or space.
• Expected Value: The weighted average of all possible values of a random variable.

Real World Examples

• Drawing cards from a deck: Determining if events like drawing a spade and drawing a heart are disjoint.
• Flipping a coin multiple times: Calculating probabilities for sequences of heads and tails (binomial).
• IRS tax return audits: Calculating the probability of multiple returns containing errors.
• Casino game payoffs: Calculating expected value to determine if a game is profitable.
• Newborn baby gender selection: Using probability tables to find the likelihood of a certain number of girls.
• Password creation: Calculating the total number of possible passwords and the probability of guessing correctly.
• Mountain rescue calls: Using the Poisson distribution to find the probability of calls per day.

People

• B. Jones: Student/Test Taker