Probability Basics Quiz

Chapter 3: Probability Concepts and Distributions

• This chapter discusses probability concepts and distributions.
• Probability is the likelihood of an outcome occurring.
• Probabilities range from 0 to 1, with 1 representing certainty.
• Outcomes are the results of a process.
• Experiments are processes generating outcomes.
• The sample space consists of all possible outcomes of an experiment.

Three Views of Probability

• Classical: Based on theory. Probability = (Favorable Outcomes) / (Total Possible Outcomes).
• Relative Frequency: Based on empirical data. Probability = (Times Event Occurred) / (Total Observations).
• Subjective: Based on judgment. Evaluates the likelihood of an event based on opinion and experience.

Basic Probability Rules and Formulas

• Probability of any outcome is between 0 and 1.
• The sum of probabilities for all outcomes in a sample space equals 1.
• The probability of the complement of event A is 1 minus the probability of A.

Events

• An event is a collection of one or more outcomes.
• Examples include rolling dice, weather conditions, and market surveys.
• The complement of an event contains outcomes not included in the event.

Rule 1

• The probability of an event is the sum of the probabilities of its constituent outcomes.

Rule 2

• The probability of the complement of an event A is equal to 1 minus the probability of A (P(A')).

Mutually Exclusive Events

• Mutually exclusive events share no common outcomes.
• The probability of either event A or event B happening is the sum of their individual probabilities (P(A or B)=P(A) + P(B)).

Rule 3

• If events A and B are mutually exclusive, then the probability that either event A or event B occurs is the sum of their individual probabilities.

Rule 4

• If events A and B are not mutually exclusive, the probability that either A or B occurs is P(A) + P(B) - P(A and B).

Conditional Probability

• Conditional probability is the probability of event A occurring, given that event B has already occurred.
• P(A|B) = P(A and B)/P(B)

Random Variables

• A numerical description of the outcome of an experiment.
• Discrete: Possible values can be counted (heads/tails).
• Continuous: Possible values fall within a range (stock prices).

Probability Distributions

• A probability distribution characterizes the values a random variable can assume and their associated probabilities.
• Described for both discrete and continuous random variables.

Discrete Random Variables

• Probability mass function (f(x)): Specifies the probability of each discrete outcome (0 ≤ f(x₁) ≤ 1 and ∑f(x) = 1).

Cumulative Distribution Function (F(x))

• The probability that the random variable X will be less than or equal to x (P(X ≤ x)).
• F(x) is the area under the probability density function to the left of x.

Continuous Probability Distributions

• Probability density function (f(x)): Describes the probability of a continuous random variable (x). A histogram of sample data can approximate the density function.

Properties of Probability Density Functions

• f(x) is always greater than or equal to zero
• The total area under the probability density function equals 1
• The probability of a single point is zero (P(X=x) = 0)
• Probabilities are defined over intervals (P(a ≤ X ≤ b), P(< c) or P(X> d)).

Other Useful Distributions

• The document lists various distributions like Bernoulli, Binomial, Poisson, Lognormal, Gamma, Weibull, Beta, Geometric, Negative Binomial, Hypergeometric, Logistic, Pareto, and Extreme Value.

Uniform Distribution

• A continuous distribution with a constant probability density between specific lower and upper limits.

Normal Distribution

• A bell-shaped, symmetric probability distribution characterized by its mean and variance.

Standardized Normal Distribution

• Special case of the normal distribution with a mean of 0 and a variance of 1.

Excel Functions

• The document includes formulas for various probability distributions (BINOM.DIST, NORM.DIST, NORM.S.DIST, POISSON.DIST).

Exponential Distribution

• Models events occurring randomly over time.

Joint and Marginal Probability Distributions

• A joint probability distribution describes the probabilities of the outcomes of two random variables occurring simultaneously. A marginal probability is associated with the outcomes of a single random variable regardless of the value of the other random variable.