Notes, summaries, assignments, exams, and problems for Mathematics

You requested the full list of major probability distributions used in computational statistics, machine learning, and data science. Below is a classification with key examples.

Types of Probability Distributions

1. Discrete Distributions (Countable Outcomes)

• Bernoulli Distribution: Binary outcome (0 or 1, e.g., a coin toss).
• Binomial Distribution: Number of successes in n independent trials.
• Negative Binomial Distribution: Number of trials required to achieve k successes.
• Geometric Distribution: Number of trials until the first success.
• Poisson Distribution: Number of events occurring in a fixed interval of time or space.
• Multinomial Distribution: Generalization of the binomial distribution for multiple categories.
• Discrete Uniform Distribution: Each

Discrete Random Variables

Discrete random variables are variables that can take on a finite number of distinct values. In simpler terms, a discrete random variable is a set of possible outcomes that is countable.

Continuous Random Variables

Continuous random variables are random variables that take an infinitely uncountable number of potential values, typically measurable amounts.

Example

1. List the sample space in the given experiment. How many outcomes are possible?

The sample space is: S = {NNN, NND, NDN, NDD, DNN, DND, DDN, DDD}

2. Count the number of defective keyboards in each outcome in the sample space and assign this number to the outcome. For instance, if you list NND, then the number of defective keyboards is 1.

The possible values of X are 0,... Continue reading "Introduction to Statistics: Discrete and Continuous Random Variables, Probability Distributions, and Sampling Techniques" »

Why Reinforcement Learning?

Reinforcement Learning (RL) is important because it enables machines to learn optimal behaviors through interaction with their environment, without needing labeled input/output pairs. It is especially useful in scenarios where the best actions are not immediately known, such as game playing, robotics, or dynamic pricing.

In RL, the agent gradually learns to take actions that maximize cumulative future rewards. Unlike supervised learning, RL focuses on long-term outcomes, rather than just immediate correctness.

Main Elements of Reinforcement Learning

• Agent: The learner or decision-maker.
• Environment: Everything the agent interacts with.
• State (S): The current situation of the environment.
• Action (A): Choices available to

Inventory Fundamentals

One use of inventory is to provide a hedge against inflation. ABC analysis divides an organization's on-hand inventory into three classes based upon annual dollar volume. Cycle counting is a process by which inventory records are verified. The difference(s) between the basic EOQ model and the production order quantity model is that the production order quantity model does not require the assumption of instantaneous delivery. Extra units that are held in inventory to reduce stockouts are called safety stock. Inventory record accuracy would be decreased by increasing stockroom accessibility. The two most important inventory-based questions answered by the typical inventory model are when to place an order and how many of

Personal Finance Math and Interest Calculations

Interest and Yield Calculations

1. Simple Interest: Katie invests $2,300 in an account that pays 7% simple interest annually. Find the future value of the account after 9 years. Round your answer to the nearest cent.
2. Monthly Compounding: Suppose you invest $1,900 at a fixed rate of 5% per year, compounded monthly. Find the future value of the account after 6 years. Round your answer to the nearest cent.
3. Continuous Compounding: Suppose you instead invest your $1,900 in an account that earns 6% interest compounded continuously. What is the total amount of your investment after 7 years? Round your answer to the nearest cent.
4. Effective Annual Yield: Find the effective annual yield to the nearest hundredth

Chapter 1: Foundations of Statistics

Data: Information derived from observations, counts, measurements, or responses.

Statistics: The science of collecting, organizing, analyzing, and interpreting data to make informed decisions.

Population: The collection of all outcomes, responses, measurements, or counts of interest.

Sample: A subset or part of a population.

Parameter: A numerical description of a population characteristic.

Statistic: A numerical description of a sample characteristic.

Descriptive Statistics: Methods to organize, display, and summarize data (e.g., mean, range, graphs, tables).

Inferential Statistics: Using sample data to draw conclusions about a population.

Qualitative Data: Attributes, labels, or non-numerical entries.

Quantitative

Introduction: A Data Science Journey

My name is Amit Kadam, and I currently reside in Mumbai. I completed my Bachelor of Engineering (B.E.) degree in 2021. After graduation, the pandemic limited job opportunities, and my family faced financial challenges, so I took my first opportunity at Sterling as a Senior Associate, where I worked for 2.5 years.

Initially, I was responsible for document verification, but I was soon promoted to manage drug health screening processes. In this role, I handled candidate health reports, prepared data for analysis, and developed strong attention to detail and data-handling skills.

During this time, a friend who successfully transitioned into data science encouraged me to explore the field. I started by learning... Continue reading "Data Science Career Transition & Predictive Modeling" »

Question Bank #1 – Net Value Functions

L03 – Engineering Economics & Net Value Applications

Review Questions

Recall the nanoRIMS example discussed in lecture. If the net value of buying the nanoparticles is $0 (the reference), determine the net value per week of having a grad student make the nanoparticles based on the following information:

• Benefit = $896/week
• Cost:
  • Cost of consumable supplies per week: Ingredients & electricity to make one batch as accurately as a grad student does is $5/100 mL * 200 mL/week = $10/week
  • Cost of time: Grad student time is $15/hr * 9 hours/100 mL * 200 mL/week = $270/week
  • Cost of space: Occupying a whole fume hood space for 16 hours during working time is $12.50/hr * 16 hrs/week = $200/week
  • Cost of any device:

What is a Scatter Diagram?

Definition

A scatter diagram (or scatter plot) is a graphical representation of two variables where each point represents an observation consisting of paired values from two datasets. The horizontal axis (X-axis) represents one variable, and the vertical axis (Y-axis) represents the other.

Construction

Each point (x_i, y_i) is plotted on the graph for the corresponding values of the two variables.

Utility in Correlation Analysis

Scatter diagrams are essential for:

• Visualizing relationships: Helps identify if a linear or non-linear relationship exists.
• Direction of correlation:
  • Positive correlation: As X increases, Y increases (points slope upwards).
  • Negative correlation: As X increases, Y decreases (points slope downwards).

14. Obtain the regression equation of Y onX and correlation coefficient for the following: X; 4 6 8 10 12 , f ;7 9 8 12 15 1. Calculate the necessary sums: X Y X² XY 2 10 4 20 3 9 9 27 7 11 49 77 8 8 64 64 10 12 100 120 ΣX = 30 ΣY = 50 ΣX² = 226 ΣXY = 308 Export to Sheets 2. Calculate the slope (b): b = (nΣXY - ΣXΣY) / (nΣX² - (ΣX)²) where n is the number of data points (n = 5 in this case) b = (5 * 308 - 30 * 50) / (5 * 226 - 30²) b = (1540 - 1500) / (1130 - 900) b = 40 / 230 b ≈ 0.174 3. Calculate the y-intercept (a): a = (ΣY - bΣX) / n a = (50 - 0.174 * 30) / 5 a = (50 - 5.22) / 5 a ≈ 8.956 4. The fitted line: Substitute the values of a and b into the equation Y = a + bX: Y = 8.956 + 0.174X Therefore, the fitted straight... Continue reading "Regression Equation and Probability Addition Theorem Solutions" »