Uniform and normal distributions are key players in continuous probability. Uniform distribution spreads probability evenly over a range, while normal distribution creates the famous bell curve. These models help us understand and predict real-world phenomena.

Both distributions have unique properties and applications. Uniform distribution is great for simulating random events, while normal distribution is crucial in statistics and data analysis. Understanding these distributions is essential for tackling probability problems and interpreting data.

Uniform Distribution

Characteristics and Functions of Uniform Distribution

• Continuous uniform distribution models random variables with constant probability over a finite interval
• Probability density function (PDF) for uniform distribution represented by a constant value within the interval [a, b]
• PDF formula for uniform distribution: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$, and 0 otherwise
• Cumulative distribution function (CDF) calculates probability of a value falling below a certain point
• CDF formula for uniform distribution: $F(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$
• Uniform distribution applications include modeling random number generators and simulating various scenarios (dice rolls)

Properties and Calculations of Uniform Distribution

• Expected value (mean) of uniform distribution: $E(X) = \frac{a+b}{2}$
• Variance of uniform distribution: $Var(X) = \frac{(b-a)^2}{12}$
• Standard deviation of uniform distribution: $\sigma = \frac{b-a}{\sqrt{12}}$
• Probability of an event occurring within a specific range [c, d] calculated using the CDF: $P(c \leq X \leq d) = F(d) - F(c)$
• Uniform distribution exhibits constant probability density, resulting in a rectangular shape when graphed

Normal Distribution Basics

Fundamentals of Normal Distribution

• Normal distribution characterized by its bell-shaped curve and symmetry around the mean
• Probability density function (PDF) for normal distribution: $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$
• Two parameters define normal distribution: mean (μ) and standard deviation (σ)
• Mean (μ) determines the center of the distribution
• Standard deviation (σ) influences the spread or width of the distribution
• Normal distribution widely used in various fields (biology, finance, social sciences)

Standard Normal Distribution and Z-scores

• Standard normal distribution represents a special case of normal distribution with mean μ = 0 and standard deviation σ = 1
• Z-score measures the number of standard deviations a data point is from the mean
• Z-score formula: $Z = \frac{X - \mu}{\sigma}$
• Z-scores allow comparison of values from different normal distributions
• Positive z-scores indicate values above the mean, negative z-scores indicate values below the mean
• Z-score table used to find probabilities associated with specific z-scores

Normal Distribution Properties

Empirical Rule and Probability Calculations

• Empirical rule (68-95-99.7 rule) describes probability distribution in normal distributions
• Approximately 68% of data falls within one standard deviation of the mean
• Approximately 95% of data falls within two standard deviations of the mean
• Approximately 99.7% of data falls within three standard deviations of the mean
• Empirical rule aids in quick probability estimations and outlier identification
• Probabilities for specific ranges calculated using z-scores and standard normal distribution table

Central Limit Theorem and Its Applications

• Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases
• Applies regardless of the underlying population distribution, given a sufficiently large sample size
• Sample size of 30 or more generally considered sufficient for Central Limit Theorem to apply
• Central Limit Theorem enables statistical inference and hypothesis testing
• Facilitates construction of confidence intervals for population parameters

Normalization and Standardization Techniques

• Normalization transforms data to a common scale while preserving the relative differences
• Min-max normalization scales values to a fixed range (0 to 1): $x_{normalized} = \frac{x - x_{min}}{x_{max} - x_{min}}$
• Z-score standardization transforms data to have mean 0 and standard deviation 1
• Standardization formula: $x_{standardized} = \frac{x - \mu}{\sigma}$
• Normalization and standardization crucial for comparing variables with different scales or units
• Applications in machine learning, data preprocessing, and statistical analysis

Assessing Normality

Q-Q Plot and Other Normality Assessment Tools

• Q-Q (Quantile-Quantile) plot graphically compares observed data distribution to expected normal distribution
• Q-Q plot construction involves plotting observed data quantiles against theoretical normal distribution quantiles
• Straight line in Q-Q plot indicates data follows a normal distribution
• Deviations from straight line suggest non-normality (skewness, heavy tails)
• Other normality assessment tools include Shapiro-Wilk test, Anderson-Darling test, and Kolmogorov-Smirnov test
• Histogram and box plot visualizations complement Q-Q plots in assessing normality
• Skewness and kurtosis measures provide numerical indicators of normality