IEE 475: Lecture D2 (2026-09-22): Probabilistic Models

Overview

Ted Pavlic builds the probability vocabulary used to create input models for discrete-event simulations, distinguishing PMFs, PDFs, CDFs, expected values, and measures of spread. He then gives practical selection rules for common distributions: use uniform or triangular models for bounded values, normal models when mean and spread are known, and exponential models for nonnegative durations with a known mean or constant hazard rate; Erlang, Weibull, and Poisson models extend these ideas.

Key takeaways

A PDF’s height is not a probability: a uniform density of 2 on [0, 0.5] is valid because its area is 1.
CDFs provide one framework for both discrete and continuous variables: they are nondecreasing, range from 0 to 1, and give interval probabilities by subtraction.
When only bounds are known, use a uniform model; when bounds and a most-likely value are known, use a triangular model.
When an unbounded variable is specified by its mean and standard deviation, a normal distribution is the maximum-entropy choice; sums of independent stages also tend toward normality.
For a nonnegative duration with a known mean and no known upper bound, an exponential model is appropriate when a constant hazard rate or memoryless behavior is plausible.
Poisson counts and exponential interarrival times describe connected views of an arrival process: one counts events per period, while the other models the time between them.

Chapters

0:00 Arena Labs, Software Access, and Textbook Version Differences

Labs 5–10 use Arena; later lab sessions are reserved for final-project work.
Arena is available on campus lab computers, through Windows installations such as Parallels or VMware, and through Canvas’s Aporto virtual desktop.
Arena versions before 16.1 use an older interface; textbook screenshots may differ, but the old features remain in the newer ribbon interface.

4:00 Random-Number Homework, Submission Expectations, and Midterm Format

In the C2 random-number assignment, each modulus remainder becomes the next seed; the next iteration must use the value after the modulus.
Homework submissions should consolidate work into one readable document; C2’s first five random-number calculations must show the hand-worked formula steps.
The midterm has two stages: a proctored LockDown Browser exam on Tuesday and an open-notes, open-collaboration stage on Thursday.

12:00 Midterm Practice Resources and Arena Competition

The Canvas midterm review can be attempted repeatedly, draws new randomized questions each time, and keeps the highest score.
Completing the LockDown Browser compliance check unlocks the midterm module, which contains sample exams from the previous eight years.
The optional Arena competition entry is due October 3; students should confirm Arena access before lab deadlines.

13:15 Stochastic Input Models and Random-Variable Ranges

Discrete-event simulations represent variation with probability distributions, even when real-world events are deterministic but impractical to reproduce step by step.
An input model is a probability distribution used as a surrogate for observed system behavior, such as interarrival times in a queue.
A random variable maps real-world outcomes to numbers; its range determines whether it is discrete, with a countable set of values, or continuous.

16:30 PMFs, PDFs, and Why Density Can Exceed One

A discrete random variable assigns probability mass to individual outcomes through a probability mass function (PMF); all outcome probabilities sum to one.
A continuous variable uses a probability density function (PDF), whose area—not height—over an interval gives probability.
PDF values may exceed one because density has units of probability per unit of x; for example, a uniform distribution on [0, 0.5] has density 2 and total area 1.

21:30 CDFs Unify Discrete and Continuous Probability Models

The cumulative distribution function is F(x) = P(X ≤ x) for both discrete and continuous random variables.
For a discrete variable, cumulative probabilities are added; for a continuous variable, density is integrated from negative infinity to x.
A valid CDF is nondecreasing, begins at 0, and approaches 1; interval probabilities are calculated as F(upper) − F(lower).

27:00 Reading Discrete CDFs and Using the Inverse-Transform Method

For service times of 3, 6, or 10 minutes with probabilities 0.30, 0.45, and 0.25, the CDF steps to 0.30, 0.75, and 1.00.
A discrete CDF is defined between its outcome values as well as at them; for example, F(7) remains 0.75 in this service-time example.
The inverse-transform method maps a uniform random number on [0, 1] through an inverse CDF to generate samples with the desired probabilities.

31:50 Continuous CDFs and Generating Exponential Random Values

An exponential CDF rises quickly near zero and approaches 1 asymptotically, reflecting many short durations and fewer long ones.
Inverting the exponential CDF converts uniform random draws from Excel, MATLAB, or R into exponentially distributed samples.
CDF-based views avoid histogram bin-size choices and also provide a direct route to calculating probabilities and generating simulation inputs.

38:00 Expected Value, Variance, and the Center-of-Mass Analogy

The mean or expected value is a distribution’s balance point; an average or sample mean is calculated from observed data.
The center-of-mass analogy explains why the mean provides a useful single-point summary of a distribution, while still discarding its variation.
Variance is the expected squared distance from the mean, and standard deviation is the square root of variance; higher moments also describe features such as skew.

51:00 Distinguishing PDFs, CDFs, and Inverse CDFs by Their Shapes

A PDF must be nonnegative and have total area 1; it can rise above 1, so height alone cannot disqualify it.
A CDF must stay between 0 and 1, be nondecreasing, and run from 0 to 1; negative x-values are allowed because x represents outcomes, not probabilities.
An inverse CDF maps probabilities in [0, 1] to outcome values; a graph that assigns multiple outputs to one input is not a function and cannot be a PDF or CDF.

54:45 Uniform Distribution for Bounded Values with No Further Information

If stakeholders provide only lower and upper bounds a and b, the uniform distribution is a low-bias default and the maximum-entropy choice under those constraints.
Uniform outcomes do not favor any point within the interval [a, b].
The probability of falling in a subinterval equals its length divided by the full range; an interval of length 2.5 within a range of 10 has probability 0.25.

57:10 Triangular Distribution for Bounds and a Most-Likely Value

Use a triangular distribution when stakeholders provide a minimum a, maximum b, and most-likely value c.
The density rises toward c on the left and falls away from c on the right, encoding the stated mode without adding an unsupported shape assumption.
In the CDF, the inflection point marks the mode; multiple inflection points can signal a multimodal distribution.

1:00:35 Normal Distribution for Known Mean, Spread, and Additive Processes

When an unbounded variable is described by its mean and standard deviation, the normal distribution is the maximum-entropy model given those values.
The central limit theorem explains why sums of independent stages tend toward a normal shape; a sequence of roughly 20 or more stages is likely to be close to normal.
A normal model can approximate positive quantities when the mean is many standard deviations above zero; a truncated normal is another option when negative values must be excluded.

1:06:35 Exponential, Erlang, Weibull, and Poisson Arrival Models

For a nonnegative variable with a known mean but no stated upper bound, the exponential distribution is a practical default; it also models waiting time under a constant hazard rate.
Exponential waiting times are memoryless: elapsed waiting time does not change the remaining-wait distribution, unlike scheduled bus arrivals that become more imminent over time.
An Erlang-k variable is a sum of k exponential stages and approaches a normal shape as k increases; the Weibull adds a shape parameter for flexible reliability and failure-time models.
Poisson distributions model discrete arrival counts over a period, while exponential distributions model the associated interarrival times; the next lecture introduces Bernoulli-based models.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Ted Pavlic.