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Statistical Techniques Using R - II - Coggle Diagram
Statistical Techniques Using R - II
Probability Basics
Probability
Description
Measure the likelihood of an event occurring, ranging from 0 {impossible} to 2 {certain}
Example
Probability of a rolling a 3 on die = 1/6
Formula
P(A)
: Probability of event A
Grouped Probability
Description
Probability distribution for grouped data, often used with intervals
Formula
fi
: Frequency of group
i
,
N
: total frequency
Random Experiment
Description
An experiment where all possible outcomes are known, but the specific outcome cannot be predicted
Characteristics
Unpredictability
Reproducibility
Multiple possible outcomes
Sample Space
Description
The set of all the possible outcomes of an experiment
Example
For a coin toss, Sample Space = {Heads, Tails}
Events and Types
Types of Events
Independent Event
The outcome of one event does not affect the other
Dependent Event
The outcome of one event affects the probability of the other
Possible & Impossible Events
Possible event has non-zero probability, while impossible has probability 0
Simple Event
An event with a single outcome
Compound Event
Combination of two or more simple event
Mutually Exclusive Event
Events that cannot happen at the same time
Equally Likely Event
Events with the same probability
Conditional Probability
Formula
P(A∣B): Probability of A given B; ∩: Intersection (both A and B occur)
Bayes' Theorem
Formula
P(A∣B): Probability of A given B, P(B∣A): Probability of B given A,
Random Variable
Random Variable
Description
A variable representing numerical outcomes of a random process
Types
Discrete (countable outcomes)
Continuous (uncountable, interval-based outcomes)
Mean Of Random Variable
Formula
μ: Mean, E(X): Expected value of X, P(x): Probability of 𝑥
Variance Of Random Variable
Formula
σ2: Variance, X: Random variable, μ: Mean
Expected Value
Formula
E(X): Expected value, x: Value, P(x): Probability of 𝑥
Probability Functions
Probability Density Function (PDF)
Description
Function representing the likelihood of a continuous random variable within a range
Probability Mass Function (PMF)
Description
Function for discrete variables representing the probability of specific values
Cumulative Distribution Function (CDF)
Description
Formula
F(x): CDF value, X: Random variable, 𝑥: Specific value
Distributions
Binomial Distribution
Formula
k: Number of successes, 𝑛: Number of trials, 𝑝: Success probability
Properties
Describes discrete events, each with two outcomes (success / failure)
Poisson Distribution
Formula
λ: Mean number of occurrences, 𝑘: Number of occurrences
Normal Distribution
Formula
𝑥: Value, 𝜇: Mean, 𝜎: Standard deviation
Standard Normal Distribution
Formula
SND
f(x): The probability density for a specific value 𝑥, x: The variable being standardized, π: Pi, approximately 3.14159, e: Euler's number, approximately 2.71828
Z-Score
Description
This formula represents the probability density function (PDF) for a standard normal distribution, which has a mean (𝜇) of 0 and a standard deviation (𝜎) of 1
Properties
Symmetrical, bell-shaped curve, mean = median = mode.
