Conditional Probability in Statistics

Definition: The probability of an event occurring given that another event has occurred.
Notation: P(A|B) reads as "the probability of A given B"
Formula: P(A|B) = P(A ∩ B) / P(B), where P(B) ≠ 0
Properties:
- P(A|B) ≥ 0
- P(A|B) ≤ 1
- P(A|B) + P(A'|B) = 1, where A' is the complement of A
Independence: Events A and B are independent if P(A|B) = P(A)
Bayes' Theorem: P(A|B) = P(B|A) * P(A) / P(B), used to update probabilities based on new information

Definition: A function that describes the probability of each possible value of a random variable
Types:
- Discrete: Assigns probabilities to specific values (e.g., binomial, Poisson)
- Continuous: Assigns probabilities to intervals of values (e.g., normal, uniform)
Properties:
- The probability of any value is between 0 and 1
- The sum of probabilities for all possible values is 1
Notable Distributions:
- Binomial: Models the number of successes in n independent trials with a constant probability of success
- Normal (Gaussian): Models continuous variables with a symmetric, bell-shaped distribution
- Poisson: Models the number of events in a fixed interval with a constant average rate

Definition: Methods for summarizing and describing the basic features of a dataset
Measures of Central Tendency:
- Mean: The average value of a dataset
- Median: The middle value of a dataset when it is sorted in order
- Mode: The most frequently occurring value in a dataset
Measures of Variability:
- Range: The difference between the largest and smallest values
- Interquartile Range (IQR): The difference between the 75th and 25th percentiles
- Variance: The average of the squared differences from the mean
- Standard Deviation: The square root of the variance
Data Visualization:
- Histograms: Graphical representations of the distribution of a dataset
- Box Plots: Graphical representations of the five-number summary (minimum, Q1, median, Q3, maximum)

The probability of an event occurring given that another event has occurred is known as conditional probability.
The notation for conditional probability is P(A|B), which reads as "the probability of A given B".
The formula to calculate conditional probability is P(A|B) = P(A ∩ B) / P(B), where P(B) ≠ 0.
Conditional probability has three properties: it is always greater than or equal to 0, less than or equal to 1, and the probability of A given B plus the probability of A' given B equals 1, where A' is the complement of A.
Two events A and B are independent if the probability of A given B is equal to the probability of A.
Bayes' Theorem is a formula used to update probabilities based on new information, and it is given by P(A|B) = P(B|A) ∗ P(A) / P(B).

A probability distribution is a function that describes the probability of each possible value of a random variable.
There are two main types of probability distributions: discrete and continuous distributions.
Discrete distributions assign probabilities to specific values, such as the binomial and Poisson distributions.
Continuous distributions assign probabilities to intervals of values, such as the normal and uniform distributions.
Probability distributions have two properties: the probability of any value is between 0 and 1, and the sum of probabilities for all possible values is 1.
Notable discrete distributions include the binomial distribution, which models the number of successes in n independent trials with a constant probability of success.
Notable continuous distributions include the normal (Gaussian) distribution, which models continuous variables with a symmetric, bell-shaped distribution, and the Poisson distribution, which models the number of events in a fixed interval with a constant average rate.

Descriptive statistics are methods for summarizing and describing the basic features of a dataset.
Measures of central tendency include the mean, median, and mode, which describe the middle or average value of a dataset.
Measures of variability include the range, interquartile range (IQR), variance, and standard deviation, which describe the spread or dispersion of a dataset.
The range is the difference between the largest and smallest values in a dataset.
The IQR is the difference between the 75th and 25th percentiles of a dataset.
The variance is the average of the squared differences from the mean of a dataset.
The standard deviation is the square root of the variance of a dataset.
Data visualization is an important aspect of descriptive statistics, and it includes graphical representations of data such as histograms and box plots.
Histograms are used to represent the distribution of a dataset, while box plots are used to represent the five-number summary (minimum, Q1, median, Q3, maximum) of a dataset.