8.2 Probability Distributions and Their Visualizations

Probability distributions are key to understanding data patterns. They help us model real-world phenomena and make predictions. Normal, binomial, and Poisson distributions are common types, each with unique characteristics and applications.

Visualizing these distributions is crucial for data analysis. Q-Q plots, probability plots, and kernel density estimation are powerful tools. They allow us to compare observed data to theoretical distributions, assess goodness-of-fit, and explore the shape of our data.

Common Probability Distributions

Normal Distribution

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• Symmetric bell-shaped curve defined by mean and standard deviation
• 68-95-99.7 rule states 68% of data falls within 1 standard deviation of mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations
• Useful for modeling continuous variables (height, weight, test scores)
• Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution

Binomial Distribution

• Models the number of successes in a fixed number of independent trials with two possible outcomes (success or failure)
• Defined by the number of trials ($n$) and the probability of success ($p$)
• Probability mass function: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$, where $k$ is the number of successes
• Mean: $\mu = np$, Variance: $\sigma^2 = np(1-p)$
• Useful for modeling discrete variables (number of defective items in a batch, number of heads in coin flips)

Poisson Distribution

• Models the number of events occurring in a fixed interval of time or space, given a known average rate and independent occurrences
• Defined by the average rate of events ($\lambda$)
• Probability mass function: $P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$, where $k$ is the number of events
• Mean and Variance: $\mu = \sigma^2 = \lambda$
• Useful for modeling rare events (number of earthquakes per year, number of customers arriving per hour)

Probability Density and Cumulative Distribution Functions

• Probability Density Function (PDF) describes the relative likelihood of a continuous random variable taking on a specific value
  • Non-negative function with the area under the curve equal to 1
  • Does not give the probability of a specific value, but the probability of falling within a range of values
• Cumulative Distribution Function (CDF) gives the probability that a random variable is less than or equal to a specific value
  • Monotonically increasing function with values between 0 and 1
  • $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$, where $f(t)$ is the PDF

Visualizing Probability Distributions

Q-Q Plot

• Quantile-Quantile plot compares the quantiles of two probability distributions
• Points plotted with theoretical quantiles on the x-axis and observed quantiles on the y-axis
• If the points fall approximately along the 45-degree reference line, the observed data follows the theoretical distribution
• Deviations from the reference line indicate differences between the distributions (heavy tails, skewness)

Probability Plot

• Similar to a Q-Q plot but compares the observed data to a specific theoretical distribution (normal, exponential, Weibull)
• Points plotted with theoretical percentiles on the x-axis and observed values on the y-axis
• If the points fall approximately along a straight line, the observed data follows the specified distribution
• Useful for assessing the goodness-of-fit and identifying outliers or deviations from the assumed distribution

Kernel Density Estimation

• Non-parametric method for estimating the probability density function of a random variable
• Constructs a smooth density curve based on the observed data points
• Each data point is represented by a kernel function (Gaussian, Epanechnikov, triangular) centered at the point
• The density estimate is the sum of the kernel functions, normalized to have an area of 1
• Bandwidth parameter controls the smoothness of the density estimate (larger bandwidth produces smoother estimates)
• Useful for visualizing the shape of the distribution without assuming a specific parametric form (multimodality, skewness, heavy tails)