PDF & CDF

         I saw that students are unclear about #PDF [probability density function] and #CDF [cumulative density function].

I would be glad to provide a comprehensive explanation of probability density functions (PDFs) and cumulative distribution functions (CDFs), incorporating the best aspects of the previous responses and addressing any potential shortcomings:

Probability Density Function (PDF): A PDF is a mathematical function that describes the probability distribution of a continuous random variable. It represents the likelihood of a random variable taking on a particular value within a certain range.

The PDF is always non-negative and its integral over its entire range must equal 1.

For a continuous random variable X, the PDF is denoted as f(x).

The probability of X falling within a certain range [a, b] is given by the integral of the PDF over that range: P(a ≤ X ≤ b) = ∫[a, b] f(x) dx.

Cumulative Distribution Function (CDF): A CDF is a mathematical function that gives the probability that a random variable is less than or equal to a certain value. It is the integral of the PDF from negative infinity to that value.

For a continuous random variable X, the CDF is denoted as F(x). The CDF is always non-decreasing and its values range from 0 to 1.

The probability of X being less than or equal to a value x is given by F(x): P(X ≤ x) = F(x).

Relationship between PDF and CDF

The PDF is the derivative of the CDF: f(x) = dF(x)/dx.

The CDF is the integral of the PDF: F(x) = ∫[-∞, x] f(t) dt.

Minimal Example

Consider the uniform distribution over the interval [0, 1].

The PDF is: f(x) = 1, 0 ≤ x ≤ 1 f(x) = 0, otherwise The CDF is: F(x) = 0, x lt; 0 F(x) = x, 0 ≤ x ≤ 1 F(x) = 1, x > 1 Key Points PDFs and CDFs are fundamental concepts in probability theory.

PDFs describe the likelihood of a random variable taking on a particular value. CDFs give the probability that a random variable is less than or equal to a certain value.

PDFs and CDFs are related through differentiation and integration.

Another small example of PDF

Given a probability density function, f(x) = 1/100, what is the probability

P(10<X<20), where X~Uniform[0, 100]?

We use the probability density function (PDF) to calculate probabilities over intervals when dealing with continuous random variables. 

Since X is uniformly distributed over [0, 100] with f(x) = 1/100,

we calculate P(10 < X < 20) as follows:

P(10 < X < 20) = ∫₁₀²₀ f(x) dx

For a uniform distribution, f(x) = 1/100:

P(10 < X < 20) = ∫₁₀²₀ (1/100) dx = 1/100 × (20 - 10) = 1/100 × 10 = 0.1

Therefore, the probability P(10 < X < 20) is 0.1.

You Can Pursue Data Science Career Even Not From Pure Mathematics Background

Certainly, several career options within the field of data science don't require advanced mathematical skills. While mathematics plays a significant role in certain aspects of data science, some roles and subfields emphasize other skills and expertise. Here are some data science career options that may be suitable for individuals with limited mathematical background:

1. Data Analyst: Data analysts primarily focus on interpreting and visualizing data to provide actionable insights. While some statistical knowledge is helpful, you don't need advanced mathematics. Proficiency in tools like Excel, SQL, and data visualization tools (e.g., Tableau, Power BI) is essential.

2. Business Intelligence Analyst: Business intelligence analysts work with data to help organizations make informed business decisions. They use data visualization tools and SQL to create reports and dashboards.

3. Data Engineer: Data engineers are responsible for collecting, storing, and maintaining data for analysis. While they need to have a good understanding of databases and data processing, advanced mathematics is not a core requirement for this role.

4. Machine Learning Engineer (in some cases): While machine learning engineers often require mathematical knowledge, some organizations prioritize practical implementation and use of machine learning frameworks over deep mathematical theory. If you focus on applying existing models and frameworks, you may not need advanced math.

5. Data Science Consultant: Data science consultants work with various clients to help them leverage data for business improvement. This role may involve more communication and problem-solving skills than advanced math.

6. Data Journalist: Data journalists analyze data to create data-driven stories and visualizations for media organizations. While data journalism requires data literacy, it doesn't typically require advanced math.

7. Data Technician: Data technicians assist with data collection, cleaning, and basic analysis tasks. These roles require attention to detail and data management skills but not advanced mathematics.

8. Data Visualization Specialist: Data visualization specialists create compelling and informative data visualizations using tools like Tableau or D3.js. This role is more about design and communication skills than advanced math.

9. Data Product Manager: Data product managers oversee the development of data-related products and services. They bridge the gap between technical teams and business stakeholders, requiring more business acumen than math expertise.

10. Data Science Trainer or Educator: If you have a passion for data science, you can pursue a career in teaching and educating others about the field. This role may involve simplifying complex concepts for learners with varying mathematical backgrounds.

While these roles may not require advanced mathematics, having a basic understanding of statistics and data analysis concepts can be beneficial. Additionally, continuously learning and upskilling in areas such as data manipulation, data visualization, and domain expertise can help you excel in these roles. Ultimately, the data science field offers a range of opportunities for individuals with diverse skills and backgrounds.

If you have a passion for storytelling and wish to combine it with data-related skills, there are career options that blend narrative and data analysis. These roles often focus on conveying insights and information in a compelling and understandable way. Here are some career options that emphasize storytelling within the data science field:

1. Data Journalist: Data journalists collect and analyze data to create data-driven stories for newspapers, magazines, online publications, and other media outlets. They use data visualization and storytelling techniques to communicate complex information to a broad audience.

2. Data Storyteller: Some organizations hire data storytellers to translate data findings into meaningful narratives that can be easily understood by non-technical stakeholders. This role involves combining data analysis skills with strong communication and storytelling abilities.

3. Data Presentation Specialist: Data presentation specialists are responsible for creating engaging and informative presentations that convey data insights. They use visuals, narratives, and storytelling techniques to make data more accessible to audiences in meetings or reports.

4. Data Visualization Designer: Data visualization designers focus on creating visually appealing and effective data visualizations. They work closely with data analysts to represent data in a way that tells a clear and compelling story.

5. Content Writer/Editor for Data-Related Content: Organizations often need content writers and editors who can write articles, blog posts, or reports related to data analysis and insights. These roles require the ability to convey technical concepts in a storytelling format.

6. Data-driven Marketing Specialist: In marketing, professionals with data skills are in demand to analyze consumer data and create marketing campaigns that tell a data-driven story. They use data insights to tailor messaging and strategies.

7. Data Communication Trainer or Educator: If you enjoy teaching and have a knack for storytelling, you can pursue a career in data communication training or education. You can help individuals or organizations improve their data storytelling skills.

8. Data Science Consultant with a Communication Focus: As a data science consultant, you can emphasize the communication aspect of your role, helping clients understand and apply data insights in their decision-making processes. Strong communication and storytelling skills are essential.

To excel in these roles, you'll need a combination of data analysis skills, storytelling abilities, and a knack for visual communication. Familiarity with data visualization tools like Tableau, storytelling techniques, and a strong understanding of the data you're working with are valuable assets. Additionally, continuous learning in both data analysis and storytelling will help you succeed in these hybrid roles where data meets narrative.

Photo by Lina Kivaka

Machine Learning - Statistics and Math Common Questions

1. What is the difference between supervised and unsupervised learning?

   - Supervised Learning: In supervised learning, the algorithm learns from labeled training data, where the input and corresponding output are provided. The goal is to learn a mapping function to make predictions on new, unseen data.

   - Unsupervised Learning: Unsupervised learning involves learning patterns and relationships from unlabeled data. It includes clustering (grouping similar data points) and dimensionality reduction (reducing the number of features while preserving important information).

2. Explain bias and variance trade-off in machine learning. 

   -  Bias:  Bias refers to the error due to overly simplistic assumptions in the learning algorithm, leading to underfitting. High bias can cause the model to miss relevant relations between features and target.

   -  Variance:  Variance is the error due to too much complexity in the model, leading to overfitting. High variance can make the model overly sensitive to noise in the training data.

 3. What is regularization? 

   - Regularization is a technique used to prevent overfitting by adding a penalty term to the loss function. It discourages the model from fitting the noise in the training data. Common regularization techniques include L1 (Lasso) and L2 (Ridge) regularization.

 4. What is the curse of dimensionality? 

   - The curse of dimensionality refers to the challenges that arise when working with high-dimensional data. As the number of dimensions increases, the data becomes sparse, and distances between points become less meaningful. This can lead to increased computation time and poor model performance.

 5. Explain ROC and AUC in the context of binary classification. 

   -  ROC Curve (Receiver Operating Characteristic):  It's a graphical representation of the performance of a binary classification model at different threshold settings. It plots the true positive rate against the false positive rate.

   -  AUC (Area Under the Curve):  AUC is the area under the ROC curve. It quantifies the model's ability to distinguish between positive and negative classes. A higher AUC indicates better model performance.

 6. What is the difference between correlation and covariance? 

   -  Covariance:  Covariance measures the degree to which two variables change together. A positive covariance indicates that as one variable increases, the other also tends to increase.

   -  Correlation:  Correlation is a standardized version of covariance that measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.

 7. What is the Central Limit Theorem? 

   - The Central Limit Theorem states that the distribution of the sample means approaches a normal distribution as the sample size increases, regardless of the original distribution of the data. This is a fundamental principle in statistics and is often used in hypothesis testing and confidence intervals.

 8. Explain gradient descent. 

   - Gradient descent is an optimization algorithm used to minimize the loss function of a machine learning model. It involves iteratively adjusting the model's parameters in the direction of the steepest descent of the loss function. The learning rate determines the step size in each iteration.

 9. What is the difference between probability and statistics? 

   -  Probability:  Probability deals with predicting the likelihood of future events based on a given set of conditions. It's used to model uncertain events.

   -  Statistics:  Statistics involves collecting, analyzing, interpreting, presenting, and organizing data. It helps us draw conclusions about the underlying population based on observed data.

 10. Explain the difference between correlation and causation. 

   -  Correlation:  Correlation indicates a statistical relationship between two variables. However, correlation does not imply a cause-and-effect relationship.

   -  Causation:  Causation implies that changes in one variable directly cause changes in another variable. Establishing causation often requires rigorous experimentation and control.

Best Way to Start Math & Stat for AI

 

unplush

As you already know that mathematics and statistics are foundation of Artificial Intelligent algorithms and concept. Let discuss how to start with them you need for AI. Here are some of the most important mathematical concepts that you will need to know:

• Linear algebra: Linear algebra is the foundation of AI. It is used to represent data, solve equations, and perform operations on data.
• Calculus: Calculus is used to understand how AI models work. It is used to find the optimal weights for machine learning models and to understand the behavior of neural networks.
• Probability and statistics: Probability and statistics are used to understand the uncertainty in data. They are used to train machine learning models and to evaluate their performance.
• Discrete mathematics: Discrete mathematics is used to deal with problems that involve discrete objects, such as sets, graphs, and trees. It is used in AI for tasks such as natural language processing and computer vision.

In addition to these mathematical concepts, you will also need to be familiar with some of the following topics:

• Algorithms: Algorithms are the steps that a computer takes to solve a problem. You will need to be familiar with a variety of algorithms, including sorting algorithms, searching algorithms, and machine learning algorithms.
• Data structures: Data structures are the way that data is organized in a computer. You will need to be familiar with a variety of data structures, such as arrays, lists, and trees.
• Programming languages: You will need to be familiar with a programming language, such as Python, Java, or C++. This will allow you to implement the algorithms and data structures that you learn about.

Let’s start with linear algebra. Linear algebra is the study of vectors, matrices, and linear transformations. It is a fundamental mathematical tool that is used in many areas of science and engineering, including AI.

Here are some of the basic concepts of linear algebra:

• Vectors: A vector is a one-dimensional array of numbers. It can be used to represent a point in space, a direction, or a velocity.
• Matrices: A matrix is a two-dimensional array of numbers. It can be used to represent a system of equations, a transformation, or a data set.
• Linear transformations: A linear transformation is a function that takes a vector and maps it to another vector. It is a fundamental concept in linear algebra and is used in many areas of AI, such as image recognition and natural language processing.

If you are new to linear algebra, I recommend that you start with the basics. Learn about vectors, matrices, and linear transformations. Once you have a good understanding of the basics, you can start learning about more advanced topics, such as eigenvalues and eigenvectors.

There are many resources available online and in libraries that can teach you linear algebra. I recommend that you find a book or online course that is well-written and easy to understand.

Here are some resources that you may find helpful:

• Linear Algebra Done Right: This book by Gilbert Strang is a classic introduction to linear algebra. It is well-written and easy to understand.
• Linear Algebra with Applications: This book by David C. Lay is another good introduction to linear algebra. It covers a wide range of topics and includes many examples.
• Khan Academy: Khan Academy has a great set of videos on linear algebra. They are free to watch and cover a wide range of topics.

Calculus is a branch of mathematics that deals with the study of change. It is divided into two main parts: differential calculus and integral calculus.

• Differential calculus is the study of how functions change. It is used to find the slope of curves, the rate of change of functions, and the area under curves.
• Integral calculus is the study of how functions accumulate. It is used to find the volume of solids, the length of curves, and the work done by forces.

Calculus is a powerful tool that is used in many different fields, including physics, engineering, economics, and statistics. It is also used in many areas of computer science, such as machine learning and artificial intelligence.

Here are some of the basic concepts of calculus:

• Functions: A function is a relationship between two sets of numbers. It can be used to represent a real-world phenomenon, such as the relationship between the temperature and the time of day.
• Limits: A limit is the value that a function approaches as its input approaches a certain value. It is used to define the derivative and integral of a function.
• Derivatives: The derivative of a function is a measure of how the function changes as its input changes. It is used to find the slope of curves, the rate of change of functions, and the area under curves.
• Integrals: The integral of a function is a measure of how the function accumulates as its input accumulates. It is used to find the volume of solids, the length of curves, and the work done by forces.

If you are new to calculus, I recommend that you start with the basics. Learn about functions, limits, derivatives, and integrals. Once you have a good understanding of the basics, you can start learning about more advanced topics, such as differential equations and vector calculus.

There are many resources available online and in libraries that can teach you calculus. I recommend that you find a book or online course that is well-written and easy to understand.

Here are some resources that you may find helpful:

• Calculus Made Easy: This book by Silvanus Thompson is a classic introduction to calculus. It is well-written and easy to understand.
• Calculus: Early Transcendentals: This book by James Stewart is another good introduction to calculus. It covers a wide range of topics and includes many examples.
• Khan Academy: Khan Academy has a great set of videos on calculus. They are free to watch and cover a wide range of topics.

Probability and statistics are two closely related branches of mathematics that deal with the analysis of data. Probability is the study of chance, while statistics is the study of how data can be collected, analyzed, and interpreted.

Probability is used to measure the likelihood of an event occurring. For example, we can use probability to calculate the likelihood of flipping a coin and getting a head, or the likelihood of rolling a dice and getting a 6.

Statistics is used to collect, organize, and analyze data. It can be used to describe data, to make inferences about populations, and to test hypotheses. For example, we can use statistics to calculate the average height of a population, or to determine whether there is a significant difference in the average height of men and women.

Probability and statistics are used in many different fields, including science, engineering, business, and medicine. They are also used in many areas of computer science, such as machine learning and artificial intelligence.

Here are some of the basic concepts of probability and statistics:

• Random variables: A random variable is a variable whose value is determined by chance. For example, the outcome of a coin flip is a random variable.
• Probability distributions: A probability distribution is a mathematical function that describes the probability of different outcomes for a random variable. For example, the probability distribution for the outcome of a coin flip is a uniform distribution.
• Sampling: Sampling is the process of selecting a subset of data from a larger population. This is often done to make inferences about the population as a whole.
• Hypothesis testing: Hypothesis testing is a statistical method for testing whether there is a significant difference between two or more groups of data.

If you are new to probability and statistics, I recommend that you start with the basics. Learn about random variables, probability distributions, sampling, and hypothesis testing. Once you have a good understanding of the basics, you can start learning about more advanced topics, such as Bayesian statistics and machine learning.

There are many resources available online and in libraries that can teach you probability and statistics. I recommend that you find a book or online course that is well-written and easy to understand.

Here are some resources that you may find helpful:

• Probability and Statistics for Engineers and Scientists: This book by Jay L. Devore is a classic introduction to probability and statistics. It is well-written and covers a wide range of topics.
• Introduction to Probability and Statistics: This book by David Freedman is another good introduction to probability and statistics. It covers a more advanced range of topics than Devore’s book.
• Khan Academy: Khan Academy has a great set of videos on probability and statistics. They are free to watch and cover a wide range of topics.

Discrete mathematics is the study of mathematical structures that can be considered “discrete” rather than “continuous”. Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in “continuous mathematics” such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term “discrete mathematics”.

Here are some of the basic concepts of discrete mathematics:

• Sets: A set is a collection of objects that are distinct from each other. Sets can be finite or infinite.
• Relations: A relation is a way of connecting two sets. For example, the relation “greater than” connects the set of all numbers to itself.
• Functions: A function is a special type of relation that assigns exactly one output to each input. For example, the function “square” assigns the square of a number to that number.
• Logic: Logic is the study of reasoning and truth. It is used to construct proofs and to analyze arguments.
• Combinatorics: Combinatorics is the study of counting. It is used to count the number of possible arrangements of objects or the number of possible solutions to problems.

Discrete mathematics is used in many different fields, including computer science, mathematics, engineering, and economics. It is also used in many areas of artificial intelligence, such as natural language processing and machine learning.

Here are some of the applications of discrete mathematics:

• Computer science: Discrete mathematics is used in computer science to design algorithms, data structures, and programming languages.
• Mathematics: Discrete mathematics is used in mathematics to study sets, functions, logic, and combinatorics.
• Engineering: Discrete mathematics is used in engineering to design circuits, networks, and systems.
• Economics: Discrete mathematics is used in economics to study games, markets, and decision making.
• Artificial intelligence: Discrete mathematics is used in artificial intelligence to design search algorithms, knowledge representation systems, and machine learning algorithms.

If you are interested in learning more about discrete mathematics, there are many resources available online and in libraries. I recommend that you find a book or online course that is well-written and easy to understand.

Here are some resources that you may find helpful:

• Discrete Mathematics and Its Applications: This book by Kenneth H. Rosen is a classic introduction to discrete mathematics. It is well-written and covers a wide range of topics.
• Discrete Mathematics with Applications: This book by Richard Johnsonbaugh is another good introduction to discrete mathematics. It covers a more advanced range of topics than Rosen’s book.
• Khan Academy: Khan Academy has a great set of videos on discrete mathematics. They are free to watch and cover a wide range of topics.

You can find out more related articles I have already written in LinkedIn and Medium.

I am a Software Architect | AI, ML, Python, Data Science, IoT, Cloud ⌨️ 👨🏽 💻

Love to learn and share knowledge to help. Thank you.

Mathematics

Statistics

Data Science

Artificial Intelligence

Probability