🃏Engineering Probability Unit 11 – Continuous Probability Distributions

Key Concepts and Definitions

• 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)
• Key terms include:
  • Support: The range of values for which the PDF is defined and non-zero
  • Parameters: Values that determine the shape and location of the distribution ($\mu$, $\sigma$)
• Continuous distributions are used to model variables that can take on any value within a range ($\mathbb{R}$, $[0,\infty)$)
• Unlike discrete distributions, probabilities for specific values are always zero $P(X=x)=0$
• Probabilities are calculated using definite integrals of the PDF over an interval $P(a \leq X \leq b) = \int_a^b f(x) dx$
• Continuous distributions are often used to approximate discrete distributions when the number of possible values is large

Types of Continuous Distributions

• Normal (Gaussian) distribution: Symmetric bell-shaped curve, characterized by mean $\mu$ and standard deviation $\sigma$
• Exponential distribution: Models time between events in a Poisson process, characterized by rate parameter $\lambda$
  • Memoryless property: The future lifetime of an item does not depend on its current age
• Uniform distribution: Equal probability over a specified interval $[a,b]$, used when all values are equally likely
• Gamma distribution: Models waiting times and is a generalization of the exponential distribution, characterized by shape $k$ and scale $\theta$
• Beta distribution: Models proportions and probabilities, characterized by shape parameters $\alpha$ and $\beta$
• Weibull distribution: Used in reliability engineering to model failure times, characterized by shape $k$ and scale $\lambda$
• Lognormal distribution: Models variables that are the product of many independent, identically distributed variables, characterized by $\mu$ and $\sigma$ of the logarithm

Probability Density Functions (PDFs)

• A PDF $f(x)$ is a function that describes the relative likelihood of a continuous random variable taking on a specific value
• Properties of a valid PDF:
  • Non-negative: $f(x) \geq 0$ for all $x$ in the support
  • Integrates to 1: $\int_{-\infty}^{\infty} f(x) dx = 1$
• The probability of a random variable falling within an interval is the area under the PDF curve over that interval
• PDFs are used to calculate probabilities, moments, and other properties of continuous distributions
• The mode of a continuous distribution is the value at which the PDF reaches its maximum
• PDFs can be transformed to create new distributions using techniques like convolution and change of variables
• The shape of the PDF determines the characteristics of the distribution (skewness, kurtosis)

Cumulative Distribution Functions (CDFs)

• A CDF $F(x)$ gives the probability that a random variable $X$ takes a value less than or equal to $x$
• Definition: $F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt$, where $f(t)$ is the PDF
• Properties of a valid CDF:
  • Non-decreasing: If $a \leq b$, then $F(a) \leq F(b)$
  • Right-continuous: $\lim_{x \to a^+} F(x) = F(a)$
  • Limits: $\lim_{x \to -\infty} F(x) = 0$ and $\lim_{x \to \infty} F(x) = 1$
• CDFs are used to calculate probabilities and quantiles (percentiles) of a distribution
• The median of a distribution is the value $x$ such that $F(x) = 0.5$
• The inverse CDF (quantile function) is used to generate random samples from a distribution
• CDFs can be used to compare different distributions and assess goodness-of-fit

Expected Value and Variance

• The expected value (mean) of a continuous random variable $X$ with PDF $f(x)$ is $E[X] = \int_{-\infty}^{\infty} x f(x) dx$
• The variance of $X$ is $Var(X) = E[(X-E[X])^2] = \int_{-\infty}^{\infty} (x-E[X])^2 f(x) dx$
  • Standard deviation is the square root of variance: $\sigma = \sqrt{Var(X)}$
• The expected value represents the average value of the random variable over a large number of trials
• Variance and standard deviation measure the spread or dispersion of the distribution around the mean
• Higher moments (skewness, kurtosis) provide additional information about the shape of the distribution
• Linearity of expectation: For constants $a$ and $b$, $E[aX+b] = aE[X] + b$
• Properties of variance: For constants $a$ and $b$, $Var(aX+b) = a^2 Var(X)$
• Covariance and correlation measure the linear relationship between two random variables

Properties and Applications

• Reproductive property: The sum of independent normally distributed random variables is also normally distributed
• Central Limit Theorem: The sum of a large number of independent random variables converges to a normal distribution
  • Allows approximation of complex distributions using the normal distribution
• Exponential distribution is the only continuous distribution with the memoryless property
• Poisson process: Events occur independently at a constant average rate, interarrival times are exponentially distributed
• Reliability engineering: Model time to failure using Weibull, exponential, or lognormal distributions
• Queuing theory: Arrival and service times are often modeled using exponential or Erlang distributions
• Extreme value theory: Model the distribution of maximum or minimum values using Gumbel, Fréchet, or Weibull distributions
• Bayesian inference: Prior and posterior distributions are often modeled using beta, gamma, or normal distributions

Transformations of Random Variables

• If $X$ is a continuous random variable with PDF $f_X(x)$ and $Y=g(X)$ is a function of $X$, then the PDF of $Y$ is given by:
  • $f_Y(y) = f_X(g^{-1}(y)) \left| \frac{d}{dy} g^{-1}(y) \right|$, where $g^{-1}$ is the inverse function of $g$
• Common transformations include:
  • Linear: $Y = aX + b$, where $a$ and $b$ are constants
  • Power: $Y = X^n$, where $n$ is a constant
  • Exponential: $Y = e^X$
  • Logarithmic: $Y = \ln(X)$
• Transformations can be used to create new distributions or simplify calculations
• The Jacobian determinant is used to account for the change in volume when transforming multivariate distributions
• Convolution is used to find the distribution of the sum of independent random variables
• Moment generating functions and characteristic functions are used to analyze transformed distributions

Practical Examples in Engineering

• Quality control: Model the distribution of product dimensions or defects using normal or Weibull distributions
• Signal processing: Noise is often modeled using Gaussian or Rayleigh distributions
  • Gaussian noise is used to model thermal noise in electronic circuits
• Hydrology: Model rainfall, river flow, and flood levels using gamma, lognormal, or Gumbel distributions
• Wind energy: Model wind speed using Weibull or Rayleigh distributions to assess power generation potential
• Material strength: Model the strength of materials using Weibull or lognormal distributions for reliability analysis
• Communication systems: Channel fading and interference are often modeled using Rayleigh, Rician, or Nakagami distributions
• Finance: Model asset returns using normal or Student's t-distributions for risk assessment and portfolio optimization
• Environmental engineering: Model pollutant concentrations using lognormal or gamma distributions for risk assessment and remediation planning