PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD ↓


Probability and Statistics


Introduction

Probability and statistics are two important branches of mathematics that are very closely intertwined. Probability is the study of randomness and uncertainty, and it provides a way to forecast the likelihood of various outcomes in uncertain situations. Statistics is the science of collecting, analyzing, interpreting, presenting, and organizing data. Together, these fields help us make predictions, test hypotheses, and make decisions based on data.

Understanding Probability

Probability measures how likely an event is to occur. It is a number between 0 and 1, where 0 indicates that an event cannot occur and 1 indicates that it is certain to occur. Probability can be expressed as:

P(Event) = Number of favorable outcomes / Total number of possible outcomes

Example of Probability

Consider the simple example of tossing a fair coin. There are two possible outcomes: heads or tails. The probability of heads is:

P(Heads) = 1 / 2 = 0.5

Similarly, the probability of getting tails is also 0.5. We can also represent it like this:

Head Tail 0.5 0.5

Basic Probability Concepts

There are several fundamental concepts in probability that you should understand:

  • Experiment: A process that leads to one or more outcomes. For example, throwing dice or drawing a card.
  • Sample space: The set of all possible outcomes from an experiment. For example, {1, 2, 3, 4, 5, 6} for a six-sided dice.
  • Event: A subset of outcomes from the sample space. It can be a single outcome or multiple outcomes.
  • Complementary Events: Events that are not part of the original event. For example, if event A is getting an even number, then the complementary event is getting an odd number. The probability of complementary events is given by:
P(A') = 1 - P(A)

Example of Complementary Events

If we throw a 6-sided dice, the probability of getting a number greater than 4 (i.e. 5 or 6) is:

P(Number > 4) = 2/6 = 1/3

Thus, the probability that the number coming out is not more than 4 is:

P(Number ≤ 4) = 1 - P(Number > 4) = 1 - 1/3 = 2/3

Conditional Probability

Conditional probability is the probability of an event occurring, provided another event has already occurred. It is represented as P(A|B), which is read as the probability of A given B.

P(A|B) = P(A ∩ B) / P(B)

Example of Conditional Probability

Suppose you have a deck of 52 cards, and you want to find the probability that a card drawn is a king, given that it is red. The probability of drawing a king (A) and the probability of the card being red (B) are:

P(King) = 4/52 = 1/13
 P(Red) = 26/52 = 1/2

Since there are 2 kings among the 26 red cards, we have:

P(King ∩ Red) = 2/52 = 1/26

Thus, the conditional probability is:

P(King | Red) = P(King ∩ Red) / P(Red) = (1/26) / (1/2) = 2/26 = 1/13

Law of Total Probability and Bayes' Theorem

Law of Total Probability

The law of total probability is used to calculate the probability of an event by considering all possible ways for the event to occur. It states that if B1, B2, ..., Bn are mutually exclusive events that form a partition of the sample space, then:

P(A) = P(A ∩ B1) + P(A ∩ B2) + ... + P(A ∩ Bn)

Using conditional probability, this can be written as:

P(A) = P(A|B1)P(B1) + P(A|B2)P(B2) + ... + P(A|Bn)P(Bn)

Bayes' theorem

Bayes' theorem is a powerful tool in probability that allows us to invert conditional probabilities. It is given as:

P(A|B) = [P(B|A) * P(A)] / P(B)

Example of Bayes Theorem

Suppose 1% of a population suffers from a particular disease and there is a test available for this disease that is 99% accurate.

  • P(disease) = 0.01 (1% have the disease)
  • P(no disease) = 0.99
  • P(positive test|disease) = 0.99
  • P(positive test|no disease) = 0.01 (false positive rate)

To find the probability that a person actually has the disease given a positive test result, use Bayes' theorem:

P(Disease|Positive Test) = [P(Positive Test|Disease) * P(Disease)] / P(Positive Test)

Where:

P(Positive Test) = P(Positive Test|Disease) * P(Disease) + P(Positive Test|No Disease) * P(No Disease)
= 0.99 * 0.01 + 0.01 * 0.99
 = 0.0099 + 0.0099
 = 0.0198

Thus, the probability that a person actually has the disease after a positive test result is:

P(Disease|Positive Test) = [0.99 * 0.01] / 0.0198 = 0.0099 / 0.0198 = 0.5

Statistics overview

As we move from probability to statistics, we focus more on data collection, analysis, and interpretation. Some of the basic concepts in statistics are as follows:

Descriptive statistics

Descriptive statistics summarize the characteristics of a data set. They can provide simple summaries about samples and measurements. Here are some key terms:

  • Mean: The average of a data set.
  • Median: The middle value when the data is sorted.
  • Mode: The value that appears most often.
  • Variance: This is a measure of how much the values in a data set vary from the mean.
  • Standard deviation: The square root of the variance, which shows how spread out the values are around the mean.

Inferential statistics

Inferential statistics allows us to make predictions or inferences about a population based on a sample of data. This includes estimating population parameters, testing hypotheses, and making predictions.

Example

Suppose we have the following data set showing the test scores of a group of 10 students:

Test Scores: 82, 90, 76, 88, 95, 79, 84, 92, 78, 81

We can calculate the mean, median and mode as follows:

  • Mean: sum of scores, divided by the number of observations:
    Mean = (82 + 90 + 76 + 88 + 95 + 79 + 84 + 92 + 78 + 81) / 10 = 84.5
  • Median: The middle score when the data is arranged in ascending order:
    Ordered Scores: 76, 78, 79, 81, 82, 84, 88, 90, 92, 95
     Median = (82 + 84) / 2 = 83
  • Mode: Most Frequent Score:
    Mode = None (all scores appear only once)

Probability distributions

Probability distributions describe how the probabilities of different outcomes are distributed in the sample space. Normal distributions include:

Discrete distribution

  • Binomial distribution: Describes the number of successes in a fixed number of independent Bernoulli trials (e.g., tossing a coin).
  • Poisson distribution: Describes the number of events that occur in a certain interval of time or space.

Continuous delivery

  • Normal distribution: Also known as the Gaussian distribution, this is a bell-shaped curve that is symmetric about the mean (e.g., people's height).
  • Exponential distribution: Describes the time between events in a Poisson process.

Conclusion

Probability and statistics together form a fundamental part of mathematics that helps understand and deal with uncertainty. From predicting outcomes based on probability models to analyzing real-world data with statistical techniques, these fields provide powerful tools for decision-making in various fields such as business, engineering, healthcare, and more. With an understanding of basic concepts such as sample space, events, probability rules, and statistical measures, one can effectively interpret data and draw conclusions that guide action. As you delve deeper into each topic, the mathematical elegance and practical applications will become apparent, demonstrating the richness and utility of probability and statistics in the real world.


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