🧰Engineering Applications of Statistics Unit 3 – Random Variables & Probability Models

Key Concepts

• Random variables assign numerical values to outcomes of random experiments
• Probability distributions describe the likelihood of different values occurring for a random variable
• Expected value represents the average value of a random variable over many trials
• Variance measures the spread or dispersion of a random variable around its expected value
  • Calculated by taking the average of the squared differences between each value and the mean
• Joint probability distributions describe the probability of two or more random variables occurring together
• Transformations of random variables involve applying functions to change their properties or create new random variables
• Random variables and probability distributions have numerous applications in engineering for modeling uncertainty and variability

Types of Random Variables

• Discrete random variables have countable values (integers, finite set of numbers)
  • Examples include the number of defective items in a batch or the number of customers arriving in an hour
• Continuous random variables can take on any value within a specified range
  • Often represented by real numbers on a continuous scale (weight, temperature, time)
• Mixed random variables have both discrete and continuous components
• Bernoulli random variables have only two possible outcomes (success or failure, 0 or 1)
  • Used to model binary events like a coin flip or a component functioning or failing
• Binomial random variables count the number of successes in a fixed number of independent Bernoulli trials
• Poisson random variables model the number of events occurring in a fixed interval of time or space
  • Useful for modeling rare events (number of defects per unit area, customer arrivals per hour)

Probability Distributions

• Probability mass functions (PMFs) define the probability of each possible value for discrete random variables
  • The sum of all probabilities in a PMF must equal 1
• Probability density functions (PDFs) describe the relative likelihood of different values for continuous random variables
  • The area under the PDF curve between two values represents the probability of the variable falling in that range
• Cumulative distribution functions (CDFs) give the probability that a random variable is less than or equal to a certain value
• Common discrete distributions include Bernoulli, binomial, Poisson, and geometric
• Common continuous distributions include uniform, normal (Gaussian), exponential, and beta
• The parameters of a distribution (mean, variance, etc.) determine its shape and properties

Expected Value and Variance

• The expected value (mean) of a discrete random variable $X$ is calculated as: $E[X] = \sum_{x} x \cdot P(X=x)$
  • Multiply each possible value by its probability and sum the results
• For a continuous random variable, the expected value is: $E[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$
• The variance of a random variable $X$ is: $Var(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2$
  • Measures how far the values are from the mean on average
• The standard deviation is the square root of the variance: $\sigma = \sqrt{Var(X)}$
• Chebyshev's inequality bounds the probability of a random variable deviating from its mean by a certain number of standard deviations
  • For any distribution, at least $1 - \frac{1}{k^2}$ of the values lie within $k$ standard deviations of the mean

Joint Probability Distributions

• Joint probability mass functions (JPMFs) give the probability of two or more discrete random variables occurring together
  • $P(X=x, Y=y)$ is the probability that $X$ takes on value $x$ and $Y$ takes on value $y$ simultaneously
• Joint probability density functions (JPDFs) describe the relative likelihood of different combinations of values for continuous random variables
• Marginal distributions are obtained by summing (discrete) or integrating (continuous) the joint distribution over the other variables
• Conditional distributions give the probability of one variable given the value of another
  • For discrete variables: $P(Y=y \mid X=x) = \frac{P(X=x, Y=y)}{P(X=x)}$
• Independence means the probability of one variable does not depend on the value of the other
  • For independent variables, the joint distribution is the product of the marginal distributions

Transformations of Random Variables

• Linear transformations involve multiplying a random variable by a constant and/or adding a constant
  • If $Y = aX + b$, then $E[Y] = aE[X] + b$ and $Var(Y) = a^2 Var(X)$
• Non-linear transformations apply a function to a random variable to create a new one
  • The PDF of the transformed variable is related to the original PDF by the Jacobian of the transformation
• Convolution is used to find the distribution of the sum of two independent random variables
  • The PDF of the sum is the convolution of the individual PDFs
• The Central Limit Theorem states that the sum of many independent random variables approaches a normal distribution
  • Useful for approximating complex distributions with the normal distribution

Applications in Engineering

• Modeling the strength of materials as random variables to assess reliability and failure probabilities
• Analyzing measurement errors and uncertainties using probability distributions
• Using the Poisson distribution to model the occurrence of rare events like equipment failures or defects
• Applying the normal distribution to describe variability in manufacturing processes and quality control
• Employing joint distributions to study the relationship between multiple variables (stress and strain, demand and supply)
• Transforming random variables to create new distributions that better fit empirical data or simplify calculations
• Utilizing the Central Limit Theorem to justify the use of normal-based statistical methods for large sample sizes

Practice Problems and Examples

• A fair six-sided die is rolled. Let $X$ be the number shown on the top face. Find the PMF, expected value, and variance of $X$.
• The time until failure of a certain component follows an exponential distribution with a mean of 10 hours. What is the probability that the component lasts more than 15 hours?
• Two fair coins are flipped. Let $X$ be the number of heads observed. Find the PMF of $X$.
  • Now let $Y$ be the number of tails observed. Find the joint PMF of $X$ and $Y$.
• The weights of apples harvested from a tree follow a normal distribution with a mean of 150 grams and a standard deviation of 20 grams. What proportion of apples weigh between 120 and 170 grams?
• A machine fills bottles with a liquid. The volume of liquid in each bottle follows a normal distribution with a mean of 500 mL and a standard deviation of 10 mL. Find the probability that the total volume in a random sample of 50 bottles exceeds 25,500 mL.
• A company produces steel cables whose breaking strengths are normally distributed with a mean of 1800 kg and a standard deviation of 150 kg. If the cables are designed to withstand a load of 1500 kg, what percentage of cables will fail under this load?