Probability & Statistics in Machine Learning

Introduction to Probability and Statistics

• Knowledge of Probability and Statistics is essential for applications in Machine Learning and various fields.
• Understanding these concepts can lead to job opportunities and research careers.

Key Concepts

• The Law of Large Numbers ensures that as more trials are conducted, the empirical probability will approach the theoretical probability.
• Empirical Probability is based on observed data, while Theoretical Probability is based on mathematical models.

Joint Probability

• Joint Probability refers to the occurrence of two events together, indicating the intersection of events.
• Example: Calculating the probability that two selected candidates are aged 18-21 without replacement.

Statistics Overview

• Statistics involves collecting, classifying, summarizing, analyzing, organizing, and interpreting data.
• Two main types of statistics are Descriptive and Inferential.

Probability Distributions

• Types of Probability Distributions include Probability Density Functions (PDFs) and Cumulative Distribution Functions (CDFs).
• The Five Number Summary includes Min, Q1, Median (Q2), Q3, and Max, providing insights into dataset distribution.

Measures of Central Tendency

• Central tendency is measured through Mean, Mode, and Median, which define the average or typical values in a dataset.

Measures of Dispersion

• Dispersion is analyzed using Standard Deviation and Variance, which show how data points spread around the central tendency.
• High variability indicates widely spread data, whereas low variability suggests data is closely clustered around the mean.

Range and Interquartile Range (IQR)

• Range is the difference between the highest and lowest values in a dataset, offering a crude measure of dispersion.
• Interquartile Range (IQR) measures the spread of the middle 50% of data, calculated as Q3 - Q1.

Variance and Standard Deviation

• Variance quantifies how much data points deviate from the mean, calculated as the average of squared deviations.
• Standard Deviation is the square root of the variance, indicating data spread in the same units as the original data.

Measures of Position

• Percentiles, Quantiles, and Standard Scores provide insights into the relative position of data points in a distribution.

Types of Random Variables

• Random Variables can be Continuous (e.g., weights within a range) or Discrete (e.g., counts of coin flips).

Random Variable Distributions

• Cumulative Probability Distribution (CDF) and Probability Density Function (PDF) explain random variable behavior.

Uniform and Gaussian Distributions

• Uniform Distribution has equal probabilities across all outcomes.
• Gaussian (Normal) Distribution is crucial for probabilities and statistics, characterized by its bell-shaped curve.

Entropy

• Entropy measures uncertainty, with zero entropy indicating no new information.
• For n equally probable messages, information conveyed = -log2(p), emphasizing distribution uniformity increases entropy.

Application Domains

• Understanding Probability and Statistics enhances analytical skills applicable across diverse domains, promoting clarity in data interpretation and decision-making.

This quiz explores the real-world applications of Probability and Statistics, particularly in the context of Machine Learning. Students will understand how these concepts can lead to career opportunities and the potential for lifelong research in the field.