Statistical Models and Distributions

Modeling and Simulation

• Statistical models are used in simulation to describe probabilistic systems, rather than deterministic ones.
• Models can be developed by sampling the phenomenon of interest.
• Selection of a distribution is made through educated guesses.
• Parameter estimations are made.
• Goodness of fit is tested.

Review of Terminology and Concepts

• Discrete random variables have a finite or countably infinite number of possible values. Examples include the number of jobs arriving at a job shop each week.
• Continuous random variables can take on any value within a given range.
• Cumulative distribution functions (CDF) show the probability that a random variable is less than or equal to a certain value (x).
• Expectation is the expected value of a random variable.
• Statistical models include discrete (Bernoulli, Binomial, Geometric, Negative Binomial, Poisson) and continuous (Uniform, Exponential, Normal, Weibull, Lognormal) distributions.

Discrete Random Variables

• The probability (p(xᵢ)) of each discrete value (xᵢ) must be greater than or equal to 0.
• The sum of all probabilities must equal 1.

Example 1 (Discrete)

• Example of tossing a single die where probability of each face is proportional to the number of spots showing.
• Probabilities of faces are 1/21, 2/21, 3/21, 4/21, 5/21, and 6/21 respectively, for faces with 1, 2, 3, 4, 5, and 6 spots.

Example 2 (Discrete)

• Illustrative graph (probability mass function) of the distribution in Example 1.

Continuous Random Variables

• The probability that a continuous random variable, X lies in the interval [a, b] is given as the integral f(x)dx from a to b.
• The probability density function (pdf), f(x), must be greater than or equal to 0 for all values of x in the range.
• The integration of the pdf over the entire range must equal 1.
• Probability of a specific value for a continuous variable is zero

Cumulative Distribution Function (CDF)

• For discrete random variables, the CDF is the sum of probabilities up to a given value (x).
• For continuous random variables, the CDF is the integral of the pdf up to a given value (x).
• Example (die rolling): Demonstrates how this looks graphically.

Expectation

• The expected value of a discrete random variable is the sum of each value multiplied by its probability.
• The expected value of a continuously distributed random variable is the integral of x times the probability density function over the entire range.

Useful Statistical Models

• Queueing Models: Queueing models deal with probabilistic interarrival and service times. Typical distributions used for interarrival and service time distributions include Poisson.
• Inventory Models: Inventory models consider variables such as demand per order/time period, time between demands, and lead times.
• Reliability and maintainability: Model time to failure (TTF), often using exponential distribution for randomly failing systems.

Discrete Distributions

• Bernoulli trials and their associated Bernoulli distribution.
• Binomial distribution (number of successes in n independent trials).
• Geometric and negative binomial distributions (number of trials until the k-th success).
• Poisson distribution.

Bernoulli Trials

• Experiments that have only two outcomes (success or failure) which are independent trials.

Binomial Distribution

• The number of successes in repeated trials.

Example Binomial

• Examples of binomial distributions, calculating probabilities for "exactly 5 heads" and "at least 5 heads" in an 8-coin toss.

Geometric Distribution

• Number of trials required to achieve a first success in a Bernoulli process.

Example Geometric

• Examples of how to apply geometric distributions.

Negative Binomial Distribution

• Number of trials until exactly k successes.

Example Negative Binomial

• Example applications of the negative binomial distribution.

Poisson Distribution

• Describes the probability of a given number of events occurring in a fixed interval of time or space. Often used for random occurrences.
• Examples: The number of calls per hour, the number of births in a year, the number of viral cases in a city, and more.

Continuous Distributions

• Uniform distribution
• Exponential distribution
• Normal distribution
• Weibull distribution
• Lognormal distribution

This quiz covers the fundamentals of statistical modeling and simulation, focusing on both discrete and continuous random variables. It includes key concepts such as parameter estimation, goodness of fit testing, and various probability distributions. Test your understanding of how these models are constructed and applied in probabilistic systems.