Lecture Notes on Probability and Decision-Making

These flashcards cover key concepts and definitions related to probability, decision-making under uncertainty, and statistical inference.

Probability

A numerical value that measures the likelihood that an event occurs.

Addition Rule

A probability rule that states the probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection.

Multiplication Rule

A probability rule that states the probability of the intersection of two events is the product of the probability of one event and the conditional probability of the other event given the first.

Bayes’ Theorem

A formula used to update probabilities based on new evidence.

Frequentist Interpretation

Assumes that probabilities represent proportions of specific events occurring over infinitely identical trials.

Bayesian Interpretation

Assumes that probabilities are subjective beliefs about the relative likelihood of events.

Experiment

A process that leads to one of several outcomes.

Outcome

The result of an experiment.

Sample Space

The set of all possible outcomes of an experiment.

Event

A subset of the sample space.

Mutually Exclusive Events

Events that do not share any common outcomes.

Exhaustive Events

Events that include all outcomes in the sample space.

Venn Diagram

A visual representation of sets and their relationships.

Complement Rule

States that the probability of an event plus the probability of its complement equals 1.

Conditional Probability

The probability of an event given that another event has occurred.

Independent Events

Events where the occurrence of one does not affect the occurrence of the other.

Variance

A measure of the dispersion of a set of values.

Random Variable

A numerical outcome from a probabilistic experiment.

Expected Value

A measure of the central tendency of a random variable.

Discrete Uniform Distribution

A distribution where each outcome is equally likely.

Binomial Distribution

Describes the number of successes in a sequence of independent Bernoulli trials.

Hypergeometric Distribution

Describes draws from a finite population without replacement.

Poisson Distribution

Estimates the number of successes over a specified interval of time or space.

Probability Mass Function (PMF)

Function that gives the probability that a discrete random variable is exactly equal to some value.

Probability Density Function (PDF)

Function that describes the likelihood of a continuous random variable taking on a particular value.

Standard Normal Distribution

A normal distribution with a mean of 0 and a standard deviation of 1.

Sampling Error

The difference between a sample statistic and the corresponding population parameter.

Statistical Inference

The process of drawing conclusions about a population based on sample data.

Central Limit Theorem (CLT)

States that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

Unbiased Estimator

An estimator whose expected value equals the true parameter being estimated.

Sampling Distribution

The probability distribution of a statistic obtained from a larger population.

Consistency

The property that ensures that as the sample size increases, the sample mean converges in probability to the population mean.

Precision

As the sample size increases, the sample mean becomes a more reliable estimate of the population mean.

Expected Value of a Random Variable

Calculated as the sum of all possible values, each multiplied by the probability of its occurrence.

Standard Error of the Mean

The standard deviation of the sampling distribution of the sample mean.

Quantiles

Values that divide a probability distribution into segments of equal probability.

Joint Probability Table

A table that displays the probability of two events occurring simultaneously.

Marginal Probability

The probability of a single event occurring without regard to other variables.

Cumulative Distribution Function (CDF)

A function that describes the probability that a random variable takes on a value less than or equal to a specific value.

Hypothesis Testing

A statistical method to determine whether a hypothesis about a population parameter is likely true.

Type I Error

The error of rejecting a true null hypothesis.

Type II Error

The error of failing to reject a false null hypothesis.

Power of a Test

The probability that it correctly rejects a false null hypothesis.

Probability

A numerical value that measures the likelihood that an event occurs. Formula: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
Example Problem: If a die is rolled, what is the probability of rolling a 4?
Solution: There is 1 favorable outcome (rolling a 4) out of 6 total outcomes. Thus, $P(4) = \frac{1}{6}$.

Addition Rule

A probability rule that states the probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection. Formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Example Problem: If P(A) = 0.3 and P(B) = 0.5, with P(A ∩ B) = 0.1, what is P(A ∪ B)?
Solution: $P(A \cup B) = 0.3 + 0.5 - 0.1 = 0.7$.

Multiplication Rule

A probability rule that states the probability of the intersection of two events is the product of the probability of one event and the conditional probability of the other event given the first. Formula: $P(A \cap B) = P(A) \times P(B \mid A)$
Example Problem: If P(A) = 0.4 and P(B \mid A) = 0.5, what is P(A ∩ B)?
Solution: $P(A \cap B) = 0.4 \times 0.5 = 0.2$.

Conditional Probability

The probability of an event given that another event has occurred. Formula: $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$
Example Problem: If P(A ∩ B) = 0.02 and P(B) = 0.1, what is P(A | B)?
Solution: $P(A \mid B) = \frac{0.02}{0.1} = 0.2$.

Bayes’ Theorem

A formula used to update probabilities based on new evidence. Formula: $P(A \mid B) = \frac{P(B \mid A) \times P(A)}{P(B)}$
Example Problem: If P(B | A) = 0.9, P(A) = 0.3, and P(B) = 0.5, what is P(A | B)?
Solution: $P(A \mid B) = \frac{0.9 \times 0.3}{0.5} = 0.54$.

Law of Total Probability

Relates marginal probabilities to conditional probabilities with a partition of the sample space. Formula: $P(B) = \sum P(B \mid A_i) P(A_i)$ where {A_i} is a partition of the sample space.
Example Problem: If P(B | A1) = 0.2, P(B | A2) = 0.5, P(A1) = 0.4, and P(A2) = 0.6, what is P(B)?
Solution: $P(B) = (0.2 \times 0.4) + (0.5 \times 0.6) = 0.08 + 0.30 = 0.38$.

Complement Rule

States that the probability of an event plus the probability of its complement equals 1. Formula: $P(A) + P(A') = 1$
Example Problem: If P(A) = 0.7, what is P(A')?
Solution: $P(A') = 1 - 0.7 = 0.3$.

Expected Value

A measure of the central tendency of a random variable. Formula: $E(X) = \sum x_i P(x_i)$ where x_i are all possible values.
Example Problem: If there are outcomes of 2 with probability 0.3, 4 with probability 0.5, and 6 with probability 0.2, what is E(X)?
Solution: $E(X) = (2 \times 0.3) + (4 \times 0.5) + (6 \times 0.2) = 0.6 + 2.0 + 1.2 = 3.8$.

Standard Error of the Mean

The standard deviation of the sampling distribution of the sample mean. Formula: $SE = \frac{s}{\sqrt{n}}$ where $s$ is the standard deviation and $n$ is the sample size.
Example Problem: If the standard deviation s = 10 and n = 25, what is SE?
Solution: $SE = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2$.

Cumulative Distribution Function (CDF)

A function that describes the probability that a random variable takes on a value less than or equal to a specific value. Formula: $F(x) = P(X \leq x)$
Example Problem: If a random variable X is distributed uniformly between 0 and 1, what is F(0.5)?
Solution: For a uniform distribution, $F(0.5) = 0.5$ (50% of the distribution lies below 0.5).

Power of a Test

The probability that it correctly rejects a false null hypothesis. Formula: $\text{Power} = 1 - P(\text{Type II error})$
Example Problem: If the probability of a Type II error is 0.2, what is the power of the test?
Solution: $\text{Power} = 1 - 0.2 = 0.8$.