Study Notes: The Normal Distribution and Financial Mathematics

Normal Distribution and Probability

The Normal Distribution

The normal distribution is a fundamental concept in probability and statistics, describing how data values are distributed around a mean. It is characterized by its bell-shaped curve and is widely used to model real-world phenomena.

• Definition: The normal distribution is a continuous probability distribution defined by its mean ($\mu$) and standard deviation ($\sigma$).

• Properties:

  • Symmetrical about the mean

  • Mean, median, and mode are equal

  • Approximately 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three (the Empirical Rule)

• Probability Density Function:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

• Applications: Used in fields such as psychology, finance, and natural sciences to model test scores, stock prices, and measurement errors.

• Example: Heights of adult humans are often normally distributed.

Problem Solving with the Normal Distribution

Solving problems with the normal distribution involves finding probabilities and percentiles using the mean and standard deviation.

• Z-score: Standardizes a value to show how many standard deviations it is from the mean.

$z = \frac{x - \mu}{\sigma}$

• Using Z-tables: Z-tables provide the probability that a value is less than a given z-score.

• Example: If $\mu = 100$, $\sigma = 15$, what is the probability that $x = 115$ or less?

• Solution: $z = \frac{115 - 100}{15} = 1$. Use Z-table to find probability.

Financial Mathematics

Percent, Sales Tax, and Discounts

Understanding percentages, sales tax, and discounts is essential for making informed financial decisions.

• Percent: A way to express a number as a fraction of 100.

• Sales Tax: A percentage added to the purchase price of goods and services.

• Discount: A reduction from the original price, usually expressed as a percentage.

• Formula for Sales Tax:

$\text{Sales Tax} = \text{Purchase Price} \times \text{Tax Rate}$

• Formula for Discount:

$\text{Discount Amount} = \text{Original Price} \times \text{Discount Rate}$

• Example: If an item costs $50 and the sales tax is 8%, the total cost is $50 + (0.08 \times 50) = $54.

Income Tax

Income tax is a government levy on individual earnings, calculated using tax brackets and rates.

• Tax Bracket: Income ranges taxed at specific rates.

• Formula:

$\text{Tax Owed} = \text{Tax Rate} \times \text{Taxable Income}$

• Example: If taxable income is $30,000 and the tax rate is 10%, tax owed is $3,000.

Simple Interest

Simple interest is calculated on the original principal only, making it straightforward to compute.

• Formula:

$I = P \times r \times t$

• Where:

  • $I$ = Interest earned

  • $P$ = Principal (initial amount)

  • $r$ = Interest rate (decimal form)

  • $t$ = Time (years)

• Example: $P = $1,000,$r = 0.05$,$t = 2$years:$I = 1,000 \times 0.05 \times 2 = $100$

Compound Interest

Compound interest is calculated on the principal and the accumulated interest, leading to exponential growth over time.

• Formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

• Where:

  • $A$ = Final amount

  • $P$ = Principal

  • $r$ = Annual interest rate (decimal)

  • $n$ = Number of times interest is compounded per year

  • $t$ = Number of years

• Example: $P = $1,000,$r = 0.05$,$n = 4$(quarterly),$t = 2$years:$A = 1,000 \left(1 + \frac{0.05}{4}\right)^{4 \times 2}$

Comparison Table: Simple vs. Compound Interest

Type	Formula	Interest Calculation	Growth
Simple Interest	$I = P \times r \times t$	On principal only	Linear
Compound Interest	$A = P \left(1 + \frac{r}{n}\right)^{nt}$	On principal and accumulated interest	Exponential

Additional info: These topics are foundational for Liberal Arts Math, covering probability, statistics, and financial mathematics, which are essential for quantitative reasoning and real-world applications.