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

1. New Followers

Formula: New followers = Final followers - Initial followers

Use it when: The case provides follower counts at the beginning and end of the year.

Meaning: Indicates the total number of new followers generated by the campaign.

2. Cost per New Follower

Formula: Cost per new follower = Campaign cost / New followers

Meaning: Shows the acquisition cost for a single new follower.

Interpretation: A lower cost per follower indicates higher campaign efficiency.

3. New Customers from Followers

Formula: New customers = New followers × Conversion rate

Meaning: Calculates how many followers converted into actual customers.

4. Customer Acquisition Cost (CAC)

Formula: CAC = Campaign cost / New customers

Interpretation: A lower CAC is preferred as it... Continue reading "Essential Marketing Metrics and Profitability Formulas" »

Small-Scale vs. Large-Scale Risk Aversion

The core idea is to understand the differences between how small and large changes in wealth affect risky gambles.

Diminishing marginal utility (risk aversion) primarily applies to large-scale gambles. This is because the utility function is sufficiently concave over lifetime changes in wealth. This concavity results in a higher utility for taking a certain outcome than for taking a gamble, even if the gamble has a higher expected return.

However, for small-scale gambles, the utility function is locally linear, yielding almost risk-neutral behavior. For wealthy individuals, the utility function is very weakly concave, leading to an asymptotically linear curvature. Thus, diminishing marginal utility cannot... Continue reading "Key Concepts in Behavioral Economics and Decision-Making" »

Role of Regression in Exploratory Data Analysis (EDA)

Regression analysis in EDA models the relationship between a dependent variable (Y) and one or more independent variables (X).

• Relationship Visualization: It helps visualize how variables interact. Fitting a line (y= ax + b) through a scatter plot identifies if the relationship is linear or non-linear.
• Correlation Identification: It identifies the nature of the association:
  • Positive Correlation: As X increases, Y increases.
  • Negative Correlation: As X increases, Y decreases.
  • No Correlation: Random distribution of points.
• Prediction: It allows for the prediction of continuous values (e.g., house prices, temperature) based on the established trend line.
• Outlier Detection: Plotting the regression line

Analyzing Algorithm Efficiency

Exponential Recursive Calls

return f(n-1) + f(n-1)

❌ Very bad algorithm
Why: It results in an exponential number of calls and massive recomputation.
Better: Use iterative multiplication or fast exponentiation.

Recursive vs. Iterative Practice

S(n) = S(n-1) + n*n*n

⚠️ Same asymptotic cost as iterative, but worse in practice
Why: Recursion adds significant stack overhead.
Better: Use a loop or a closed-form formula.

Efficient Linear Algorithms

return Q(n-1) + 2*n - 1

✅ Efficient linear algorithm

Multiplications: Θ(n), Additions: Θ(n).
Why: There is no redundant computation.

Optimal Element Inspection

temp = recursive call
if temp <= A[n-1]

✅ Optimal
Why: You must look at every element; therefore, Θ(n) is unavoidable.... Continue reading "Algorithm Efficiency and Summation Reference" »

Finding the Point of Intersection

Method 1: Elimination

1. Use multiplication to get one variable having the same numbers in both equations (they can have different signs).
2. If the signs for the variables are the same, subtract the two equations; otherwise, add them.
3. Substitute the answer into an original equation and solve.

Method 2: Substitution

1. Isolate one variable in one equation.
2. Substitute into the other equation.
3. Solve.
4. Substitute the answer into the original and solve.

Triangle Centers and Properties

Finding the Median

1. Determine the midpoint of the opposite line using the midpoint formula.
2. You now have two points; determine the slope of the median.
3. Use the slope and one point to determine the equation of the median.

Finding the Altitude

1. Determine the slope

Journal Entries: Recording Business Transactions

Format of Journal Entries

Examples of Journal Entries

1. Started Business with Cash ₹1,00,000

   • Cash A/c Dr ₹1,00,000
     To Capital A/c ₹1,00,000
   • (Being business started with cash)

2. Purchased Goods for Cash ₹20,000

   • Purchases A/c Dr ₹20,000
     To Cash A/c ₹20,000
   • (Being goods purchased for cash)

3. Sold Goods to Priya for ₹10,000 on Credit

   • Priya A/c Dr ₹10,000
     To Sales A/c ₹10,000
   • (Being goods sold to Priya on credit)

4. Paid Rent ₹5,000

   • Rent A/c Dr ₹5,000
     To Cash A/c ₹5,000
   • (Being rent paid in cash)

Ledger Posting: Classifying Transactions

Format of Ledger Accounts

Key Statistical Concepts

Understanding Percentiles

The Xth percentile means X% of the data must fall strictly below it. The percentile of X can be calculated using the formula: (# observations (N - 1) / 2 / N * 100%).

Variance: Population vs. Sample

• The sample variance is the sum of the squared deviations from the mean divided by the number of measurements minus one.
• The population variance is the sum of the squared deviations from the mean divided by the number of measurements.

The Empirical Rule

Also known as the 68-95-99.7 rule, the Empirical Rule states that for a normal distribution:

• Approximately 68% of the measurements will fall within one standard deviation of the mean.
• Approximately 95% of the measurements will fall within two standard deviations

Introduction to Statistical Methods and Examples

Initial Setup and Data Visualization

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Key Concepts in Statistical Hypothesis Testing

Statistical Hypothesis

To reach decisions about populations based on sample information, we make certain assumptions about the populations involved. Such assumptions, which may or may not be true, are called statistical hypotheses.

Null Hypothesis (H₀) and Alternative Hypothesis (H₁)

The hypothesis formulated for the purpose of its rejection, under the assumption that it is true, is called the Null Hypothesis, denoted by H₀. The hypothesis complementary to the null hypothesis is called the Alternative Hypothesis, denoted by H₁.

Test of Significance

The process that helps us decide about the acceptance... Continue reading "Statistical Hypothesis Testing and Markov Chain Problem Solutions" »

1. Emilie's 3-Year Payment Plan

Answer: 12.4%

Formula: Interest = P × r × t

• Total payment: 36 months × $33.70 = $1,213.20
• Total interest paid: $1,213.20 - $884.92 = $328.28
• Rate calculation: $328.28 = $884.92 × r × 3; r = 0.1236 or 12.4%

2. 312 Sq Ft Family Room Flooring

Answer: 22.7%

• Total cost: $1.37 × 312 = $427.44
• Tax amount: $427.44 × 0.068 = $29.07
• Total purchase: $427.44 + $29.07 = $456.51
• Total repayment: 24 × $27.66 = $663.84
• Total interest: $663.84 - $456.51 = $207.33
• Interest rate: $207.33 / ($456.51 × 2) = 0.227 or 22.7%

3. Sea Drift Motel Loan

Answer: 0.44%

Using the simple interest formula I = P × r × t:

• Interest (I): $97,000
• Principal (P): $1,000,000
• Time (t): 22 years
• Rate (r): 0.0044 or 0.44%

4. Effective Interest Rate Calculation

Using... Continue reading "Financial Mathematics: Interest and Loan Calculation Steps" »

Basic Probability Notations and Examples

Probability is the branch of mathematics that deals with the likelihood of occurrence of an event. The probability of an event ranges from 0 to 1, where:

• 0 indicates an impossible event.
• 1 indicates a certain event.

Probability is widely used in Artificial Intelligence, Machine Learning, Statistics, Data Science, and Decision Making.

1. Sample Space (S)

The sample space is the set of all possible outcomes of an experiment.

Example: When a die is rolled, S = {1, 2, 3, 4, 5, 6}.

2. Event (A)

An event is a subset of the sample space.

Example: Event A = Getting an even number, so A = {2, 4, 6}.

3. Probability of an Event

The probability of an event is calculated as:

P(A) = (Number of favorable outcomes) / (Total number