Statistics Study Guide: Probability, Random Variables, Distributions, and Combinatorics

Probability and Independence

Basic Probability Concepts

Probability is the measure of the likelihood that an event will occur. Events can be classified as independent, dependent, or mutually exclusive.

• Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other.

• Dependent Events: The occurrence of one event affects the probability of the other.

• Mutually Exclusive Events: Events that cannot occur at the same time.

• Disjoint Events: Another term for mutually exclusive events.

Example: If 55% of students live on campus, 60% have a meal program, and 33% do both, the events are not independent.

Sample Spaces and Counting

Sample Space Definition

The sample space is the set of all possible outcomes of a random experiment.

• Example: For 3 traffic lights, each can be stopped (S) or not stopped (A). The sample space is all combinations, e.g., SSS, SSA, SAS, etc.

Random Variables and Probability Distributions

Types of Random Variables

• Discrete Random Variable: Takes on countable values (e.g., number of patients showing up).

• Continuous Random Variable: Takes on any value in an interval (e.g., exam scores).

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials.

• Parameters: $n$ (number of trials), $p$ (probability of success)

• Probability Mass Function: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Mean: $\mu = np$

• Variance: $\sigma^2 = np(1-p)$

• Example: Probability all 5 patients show up if $p = 0.8$: $P(X=5) = 0.8^5$

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its mean $\mu$ and standard deviation $\sigma$.

• Probability Density Function: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

• Standard Normal: $\mu = 0$, $\sigma = 1$

• Z-score: $z = \frac{x - \mu}{\sigma}$

• Example: Exam scores with $\mu = 435$, $\sigma = 72$

Approximating Binomial with Normal

For large $n$, the binomial distribution can be approximated by the normal distribution:

• Mean: $\mu = np$

• Variance: $\sigma^2 = np(1-p)$

• Continuity Correction: Add or subtract 0.5 when approximating discrete probabilities.

Conditional Probability and Bayes' Theorem

Conditional Probability

The probability of event A given event B is:

• $P(A|B) = \frac{P(A \cap B)}{P(B)}$

Bayes' Theorem:

• $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

Combinatorics: Counting and Arrangements

Counting Principles

• Permutations: Arrangements of objects where order matters. $n!$ for $n$ distinct objects.

• Combinations: Selections where order does not matter. $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

• Example: Number of ways to choose 5 students from 12: $\binom{12}{5}$

Applications of Probability and Statistics

Medical Statistics Example

• Probability that all patients show up: Use binomial formula.

• Expected number: $E[X] = np$

Real Estate Exam Scores Example

• Percentiles: Use z-score and normal table.

• Middle 85%: Find z-scores corresponding to 7.5th and 92.5th percentiles.

Cybersecurity Analyst Example

• Probability of correct identification: Use law of total probability.

• Expected value: $E[X] = np$ for binomial process.

Standard Normal Table

Purpose and Usage

The standard normal table provides the area under the standard normal curve between the mean and a given z-score.

Z	Area
1.45	0.4265
2.00	0.4772
2.50	0.4938
3.00	0.4987

Example: For $z = 1.45$, the area between the mean and z is 0.4265.

Summary Table: Probability Distributions

Distribution	Type	Parameters	Mean	Variance
Binomial	Discrete	n, p	$np$	$np(1-p)$
Normal	Continuous	$\mu, \sigma$	$\mu$	$\sigma^2$

Additional info:

• Some questions involve using the normal approximation to the binomial for large sample sizes.

• Combinatorics questions require understanding of permutations and combinations, especially with restrictions.

• Conditional probability and independence are tested with true/false and calculation questions.

• Standard normal table is provided for z-score calculations.