🎲Intro to Probability Unit 6 – Continuous Variables & Distributions

Key Concepts

• Continuous random variables can take on any value within a specified range or interval
• Probability is determined by the area under the curve of a probability density function (PDF)
• Cumulative distribution functions (CDFs) calculate the probability of a random variable being less than or equal to a specific value
• Expected value represents the average outcome of a continuous random variable over an infinite number of trials
• Variance and standard deviation measure the spread or dispersion of a continuous probability distribution
  • Variance is the average squared deviation from the mean
  • Standard deviation is the square root of the variance
• Common continuous distributions include uniform, normal (Gaussian), exponential, and gamma distributions
• Continuous probability distributions are used to model various real-world phenomena (time between events, physical measurements)

Types of Continuous Distributions

• Uniform distribution has a constant probability density over a specified interval (rolling a fair die)
  • Probability is equal for any value within the interval
  • PDF is a horizontal line over the interval
• Normal (Gaussian) distribution is symmetric and bell-shaped, with mean $\mu$ and standard deviation $\sigma$
  • Approximately 68% of data falls within one standard deviation of the mean
  • Approximately 95% of data falls within two standard deviations of the mean
• Exponential distribution models the time between events in a Poisson process (time between customer arrivals)
  • Characterized by the rate parameter $\lambda$, which represents the average number of events per unit time
• Gamma distribution is a generalization of the exponential distribution, with shape parameter $k$ and scale parameter $\theta$
  • Models waiting times for the $k$-th event in a Poisson process
• Beta distribution is defined on the interval [0, 1] and is characterized by two shape parameters, $\alpha$ and $\beta$
  • Used to model probabilities, proportions, and percentages (success rate of a new product)

Probability Density Functions (PDFs)

• PDFs define the likelihood of a continuous random variable taking on a specific value
• The area under the PDF curve between two points represents the probability of the random variable falling within that range
• Properties of a valid PDF:
  • Non-negative: $f(x) \geq 0$ for all $x$
  • Integrates to 1: $\int_{-\infty}^{\infty} f(x) dx = 1$
• To find the probability of a random variable $X$ falling within an interval $[a, b]$, integrate the PDF over that interval: $P(a \leq X \leq b) = \int_{a}^{b} f(x) dx$
• The PDF of a continuous uniform distribution over the interval $[a, b]$ is: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$
• The PDF of a normal distribution with mean $\mu$ and standard deviation $\sigma$ is: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Cumulative Distribution Functions (CDFs)

• CDFs give the probability that a random variable $X$ is less than or equal to a specific value $x$
• Denoted as $F(x) = P(X \leq x)$
• Properties of a valid CDF:
  • Non-decreasing: If $x_1 \leq x_2$, then $F(x_1) \leq F(x_2)$
  • Right-continuous: $\lim_{x \to a^+} F(x) = F(a)$
  • Limits: $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$
• To find the probability of a random variable $X$ falling within an interval $[a, b]$, subtract the CDF values: $P(a \leq X \leq b) = F(b) - F(a)$
• The CDF of a continuous uniform distribution over the interval $[a, b]$ is: $F(x) = \frac{x-a}{b-a}$ for $a \leq x \leq b$
• The CDF of a normal distribution with mean $\mu$ and standard deviation $\sigma$ is: $F(x) = \Phi(\frac{x-\mu}{\sigma})$, where $\Phi$ is the standard normal CDF

Expected Value and Variance

• The expected value (mean) of a continuous random variable $X$ with PDF $f(x)$ is: $E[X] = \int_{-\infty}^{\infty} x f(x) dx$
  • Represents the average value of the random variable over an infinite number of trials
• The variance of a continuous random variable $X$ with PDF $f(x)$ is: $Var(X) = E[(X - E[X])^2] = \int_{-\infty}^{\infty} (x - E[X])^2 f(x) dx$
  • Measures the average squared deviation from the mean
• The standard deviation is the square root of the variance: $\sigma = \sqrt{Var(X)}$
  • Measures the spread of the distribution in the same units as the random variable
• For a continuous uniform distribution over the interval $[a, b]$, the expected value is $E[X] = \frac{a+b}{2}$, and the variance is $Var(X) = \frac{(b-a)^2}{12}$
• For a normal distribution with mean $\mu$ and standard deviation $\sigma$, the expected value is $E[X] = \mu$, and the variance is $Var(X) = \sigma^2$

Common Continuous Distributions

• Uniform distribution: Equal probability over a specified interval (random number generation)
• Normal (Gaussian) distribution: Symmetric, bell-shaped curve; models many natural phenomena (heights, weights, errors)
• Exponential distribution: Models the time between events in a Poisson process (radioactive decay, customer arrivals)
• Gamma distribution: Generalization of the exponential distribution; models waiting times (insurance claims, queue lengths)
• Beta distribution: Models probabilities, proportions, and percentages; defined on the interval [0, 1] (product quality control)
• Chi-square distribution: Models the sum of squared standard normal random variables (goodness-of-fit tests)
• Student's t-distribution: Similar to the normal distribution but with heavier tails; used for small sample sizes (confidence intervals)
• F-distribution: Ratio of two chi-square random variables; used in analysis of variance (ANOVA) tests

Applications in Real-World Scenarios

• Finance: Modeling stock prices, option pricing (Black-Scholes model), portfolio optimization
• Engineering: Reliability analysis, quality control, tolerances in manufacturing processes
• Physics: Modeling particle velocities (Maxwell-Boltzmann distribution), quantum mechanics (wave functions)
• Biology: Population dynamics, spread of diseases, genetic inheritance patterns
• Psychology: Modeling reaction times, test scores, and other human behaviors
• Meteorology: Predicting weather patterns, rainfall, and temperature distributions
• Telecommunications: Modeling signal strength, interference, and network traffic
• Operations research: Queuing theory, inventory management, and supply chain optimization

Problem-Solving Techniques

• Identify the type of continuous distribution based on the given information or context
• Determine the parameters of the distribution (mean, standard deviation, shape, scale)
• Use the PDF or CDF to calculate probabilities for specific intervals or values
  • Integrate the PDF over the desired interval
  • Subtract CDF values for the endpoints of the interval
• Apply the expected value and variance formulas to characterize the distribution's central tendency and spread
• Standardize normal distributions using z-scores: $z = \frac{x-\mu}{\sigma}$
  • Use standard normal tables or calculators to find probabilities
• Utilize moment-generating functions (MGFs) to derive the mean and variance of a distribution
• Apply the central limit theorem for large sample sizes: The sum or average of many independent random variables approaches a normal distribution
• Use simulation techniques (Monte Carlo) to approximate probabilities and expected values for complex distributions