STAT 400 Statistics and Probability I
The Complete Exam Bible · Fall 2026STAT 400 Overview
- 4 credit hours
- Fall 2026
- Statistics
- In-person paper exams
- Calculus II prerequisite
What STAT 400 covers
STAT 400 Statistics and Probability I at the University of Illinois Urbana-Champaign is an introduction to mathematical statistics that develops probability as it is needed.
- Assessed by Weekly homework, participation quizzes, three midterm exams and a final exam.
- Hardest step Naming the right model or test before any calculation starts.
- Replacement rule A final exam score above your lowest midterm replaces that midterm.
- How to prepare Rework one complete model solution per chapter before each midterm.
How STAT 400 is assessed
| Component | Weight | Format |
|---|---|---|
| Homework Assignments | 30% | 14 weekly sets submitted on Gradescope; lowest score dropped |
| Participation Assignments | 5% | Canvas quiz items plus Discussion group exercises |
| Midterm Exam 1 | 15% | In-person paper exam in lecture time |
| Midterm Exam 2 | 15% | In-person paper exam in lecture time |
| Midterm Exam 3 | 15% | In-person paper exam in lecture time |
| Final Exam | 20% | In-person paper exam at a combined time; always counts at 20% |
Weights are from the Fall 2026 syllabus and total 100%. If your final exam score is higher than your lowest midterm score, it replaces that midterm. Students must take part in at least 10 of the 14 weekly Discussions: meeting this keeps Discussion participation points and earns a second homework drop, while missing it sets all Discussion participation points to zero. Course percentages are truncated to the hundredths place before letter grades are assigned.
Assessment structure
Left to right: homework, participation assignments, Midterm Exams 1, 2 and 3, and the final exam. Segment widths reproduce the published weights and total 100%.
Current STAT 400 dates
| Date | Item | Control |
|---|---|---|
| September 25, 2026 | Midterm Exam 1 | Friday, in your regular lecture period |
| October 23, 2026 | Midterm Exam 2 | Friday, in your regular lecture period |
| November 20, 2026 | Midterm Exam 3 | Friday, in your regular lecture period |
Dates are as published in the Fall 2026 course calendar. Confirm exact deadlines and submission settings in the live LMS.
What STAT 400 covers
Eleven chapters follow the Fall 2026 calendar from the rules of probability through random variables and the central limit theorem to estimation, confidence intervals and hypothesis tests.
Probability Axioms and Set Operations
Sample spaces, set notation, the three probability properties and series-based modelsWhat this chapter sets up
STAT 400 begins with the language every later topic depends on. A random experiment has outcomes that cannot be predicted in advance; the sample space S lists all of them; an event is any subset of S.
- Sample spaces and events
- Set operations and Venn regions
- The three probability properties
- Complement and addition rules
- Inclusion-exclusion for three events
- Normalising infinite models with series
Conditional Probability, Independence and Bayes Theorem
Conditioning, probability trees, independence tests and reversing a condition with BayesConditioning changes the reference set
Conditional probability answers a different question from an ordinary probability. P(A) measures A against the whole sample space; P(A | B) measures A only among outcomes where B happened.
- Conditional probability as a ratio
- Multiplication rule and probability trees
- Law of total probability
- Independence and its complements
- Pairwise versus mutual independence
- Bayes theorem and base rates
Discrete Random Variables and Their Moments
Probability mass functions, CDFs, expected value, variance and moment generating functionsFrom outcomes to numbers
A random variable attaches a number to each outcome, which lets probability questions become algebra. A discrete random variable takes a finite or countably infinite set of values, called its support.
- Support and probability mass functions
- Cumulative distribution functions
- Expected value of a function
- Variance and the shortcut formula
- Linear transformations
- Moment generating functions
Binomial, Geometric, Poisson and Related Models
Binomial, geometric, negative binomial, Poisson, hypergeometric and multinomial modelsRecognition is the main skill
The second content block of the course is built around a family of named discrete distributions.
- Bernoulli and binomial distributions
- Geometric waiting times and memorylessness
- Negative binomial distribution
- Poisson rates and rescaling
- Hypergeometric sampling without replacement
- Multinomial counts
Continuous Random Variables and the Normal Model
Densities, percentiles, uniform, exponential and gamma models and normal probabilitiesProbability becomes area
A continuous random variable is described by a probability density function, a nonnegative curve with total area one.
- Probability density functions
- CDFs and percentiles
- Continuous expectation and variance
- Uniform and exponential models
- Gamma waiting times
- Normal distribution and z-scores
Joint Distributions, Covariance and Correlation
Joint and marginal tables, independence, covariance, correlation and linear combinationsTwo variables in one table
A joint probability mass function assigns a probability to every pair of values of two discrete random variables, and for STAT 400 it is usually displayed as a table.
- Joint probability tables
- Marginal and conditional distributions
- Independence of random variables
- Covariance and correlation
- Variance of linear combinations
Sums of Random Variables and the Central Limit Theorem
Sums of normal variables, the sample mean, the CLT and continuity correctionWhat always adds and what only sometimes survives
For any random variables, the mean of a sum is the sum of the means. For independent variables, the variance of a sum is the sum of the variances.
- Means and variances of sums
- Linear combinations of normal variables
- Distribution of the sample mean
- Central limit theorem
- Normal approximation to the binomial
- Continuity correction
Point Estimation with Likelihood and Moments
Maximum likelihood, method of moments, bias and mean square errorFrom probability to statistics
The third content block turns the direction of reasoning around. Instead of a known model predicting data, we now have data and an unknown parameter.
- Estimators and estimates
- Likelihood and log-likelihood
- Maximum likelihood estimators
- Method of moments
- Unbiasedness and bias
- Mean square error
Confidence Intervals and Sample Size
Chi-square and t distributions, intervals for means, proportions and variances, sample sizeSampling distributions come first
An interval estimate needs to know how a statistic varies from sample to sample. For a normal population three results do the work. The standardised sample mean with known σ is standard normal.
- Chi-square and t distributions
- Degrees of freedom
- Intervals for a mean with σ known or unknown
- Intervals and bounds for a proportion
- Interval for a variance
- Sample size for a target margin
Hypothesis Tests for Means, Proportions and Variances
Errors, p-values, one-sample and two-sample tests for means, proportions and variancesA test is a structured argument
The course introduces hypothesis testing through the comparison with a criminal trial. The null hypothesis plays the presumption of innocence and is kept unless the evidence against it is strong enough.
- Hypotheses, errors and significance level
- Critical values, p-values and interval evidence
- Tests for one mean
- Tests for one proportion
- Test for one variance
- Two-sample tests for means and proportions
Power and Chi-Square Tests
Power functions, sample size for power, goodness of fit, independence and homogeneityPower: how often a test finds a real effect
The power of a test is the probability of rejecting the null hypothesis when a specific alternative is true, which equals one minus the Type II error probability. Computing it takes two stages.
- Power and Type II error
- Power functions
- Sample size for a target power
- Chi-square goodness of fit
- Tests of independence
- Tests of homogeneity
Its catalog description lists the calculus of probability, random variables, expectation, distribution functions, the central limit theorem, point estimation, confidence intervals and hypothesis testing.
It is a four credit hour course with Calculus II as a prerequisite, and it serves both as a one-term introduction to statistics and as preparation for later courses such as STAT 410 and STAT 425.
How the semester is organised
The Fall 2026 calendar divides the material into three content blocks, each ending in a midterm exam.
The first block runs from basic probability through conditional probability, independence, Bayes theorem, discrete random variables, expected value, variance and moment generating functions. The second covers the named discrete distributions, continuous random variables, the uniform, exponential, gamma and normal models, joint distributions, covariance and correlation, and the central limit theorem with normal approximations.
The third turns to statistics: maximum likelihood and method of moments estimation, unbiasedness, the chi-square and t distributions, confidence intervals, sample size selection and the first hypothesis tests. The last weeks cover two-sample tests, tests for a variance, the power of a test and chi-square tests for count data.
How this guide follows the course
The eleven chapters track that sequence.
Each one opens with the definitions and results the course's learning objectives list for those weeks, works complete model solutions with numbers you can check by hand, and ends with short practice items with full answers.
Front matter explains the grading rules in detail, a mixed practice section trains the model-recognition step that exams reward, and an exam-morning chapter collects the formulas and traps worth rereading before each midterm.
What students find hardest
Three skills separate strong answers from weak ones.
The first is recognising the model: deciding whether a count is binomial, geometric, Poisson or hypergeometric before computing anything. The second is using calculus comfortably, since series, integration by parts and differentiation of a log-likelihood appear throughout. The third is writing inference conclusions in words, because a confidence interval or test decision is incomplete without a sentence about the population.
The course asks for numeric answers to three significant digits with full precision kept until the final step, and it expects work to be shown even when a calculator or R produced the number.
How the grade works
Homework and participation make up 35% of the grade and reward steady weekly work; the three midterms and the final make up the remaining 65%.
Two rules change planning: your lowest homework score is dropped, with a second drop available through the Discussion requirement, and a final exam score above your lowest midterm replaces that midterm. Course percentages are truncated rather than rounded when letter grades are assigned.
Worked example · free
From a sample proportion to a confidence interval
- 1The MLE of a Bernoulli proportion is the sample proportion: p̂ = 40/64 = 0.625.
- 1Check the large-sample condition: 40 successes and 24 failures, both comfortably above 10.
- 2Standard error = √(0.625 × 0.375 / 64) = 0.0605.
- 2Interval = 0.625 ± 1.96(0.0605) = 0.625 ± 0.119, giving (0.506, 0.744).
Key terms
- Random Experiment
- A process whose outcome cannot be predicted with certainty, described by a sample space of possible outcomes.
- Probability Distribution
- A description of the values a random variable can take and how probability is spread across them, through a pmf, pdf or CDF.
- Expectation
- The long-run average value of a random variable, computed as a probability-weighted sum or integral.
- Variance
- The expected squared deviation of a random variable from its mean, a measure of spread in squared units.
- Normal Distribution
- The symmetric bell-shaped continuous distribution set by a mean and a standard deviation, standardised with a z-score.
- Sample Mean
- The average of the observations in a random sample, with mean μ and standard error σ/√n.
- Estimator
- A rule that turns sample data into a guess for an unknown population parameter.
- Confidence Level
- The long-run proportion of intervals from a method that contain the parameter, such as 95%.
- Significance Level
- The probability of rejecting a true null hypothesis that a test is designed to allow, written α.
STAT 400 FAQ
How is the Fall 2026 grade weighted?
Homework counts 30%, participation assignments 5%, each of the three midterm exams 15%, and the final exam 20%. The weights come from the Fall 2026 syllabus and total 100%, with the lowest homework score dropped before the homework average is taken.
Can the final exam replace a midterm?
Yes. If your final exam percentage is higher than your lowest midterm percentage, the final replaces that midterm score as well as counting as the final exam at its own 20% weight.
What happens if I miss Discussion sessions?
The syllabus requires participation in at least 10 of the 14 weekly Discussions. Meeting that keeps every Discussion participation point you earned and adds a second dropped homework score; falling short sets all Discussion participation points to zero, although no minimum grade to pass the course is attached to it.
What calculator can I use in exams?
Any calculator type is allowed, ideally one that handles exponents, combinations and factorials. Each student uses one calculator, calculators cannot be shared, and phones or smartwatches cannot be used in exams.
What does the course cover?
Probability rules, conditional probability and Bayes theorem, discrete and continuous random variables, joint distributions, the central limit theorem, point estimation, confidence intervals, hypothesis tests, power and chi-square tests.
Do I need calculus for this course?
Yes. Calculus II is the prerequisite and should not be taken in the same semester. Series, integration and differentiation are used throughout, from normalising probability models to deriving maximum likelihood estimators.
How should I study for the midterms?
Study by content block. Rework complete model solutions without notes, practise identifying the right distribution or test from a word problem, and keep answers to three significant digits with the working shown.
Questions students ask in each STAT 400 chapter
- Why does the addition rule subtract the intersection?
- Outcomes in the overlap are counted once inside P(A) and again inside P(B). Subtracting P(A∩B) removes the second copy, so each outcome in the union is counted exactly once. Chapter 01 · Probability Axioms and Set Operations
- What is the difference between independent and mutually exclusive events?
- Mutually exclusive events cannot happen together, so their joint probability is zero. Independent events satisfy P(A∩B) = P(A)P(B). If both events have positive probability, being mutually exclusive forces dependence. Chapter 02 · Conditional Probability, Independence and Bayes Theorem
- Why is E[X²] not equal to the square of E[X]?
- Squaring and averaging do not commute: averaging the squares weights large values more heavily than squaring the average does. The difference E[X²] − (E[X])² is exactly the variance, which is positive whenever X is not constant. Chapter 03 · Discrete Random Variables and Their Moments
- How do I tell a binomial question from a geometric one?
- Ask what is fixed. If the number of trials is fixed and you count successes, it is binomial. If you keep going until a success and count the trials needed, it is geometric. Chapter 04 · Binomial, Geometric, Poisson and Related Models
- Why is the probability of an exact value zero for a continuous variable?
- Probability is area under the density, and a single point has no width, so its area is zero. Only intervals carry positive probability, which is also why including or excluding an endpoint never changes a continuous answer. Chapter 05 · Continuous Random Variables and the Normal Model
- Does zero correlation mean two variables are independent?
- No. Zero correlation rules out a linear trend only. A variable and its square, for a distribution symmetric about zero, have zero correlation even though one determines the other. Chapter 06 · Joint Distributions, Covariance and Correlation
- How large does n need to be for the central limit theorem?
- It depends on the population. A symmetric population can look normal at n = 5 or so, while a skewed one needs more; many courses use n of about 30 as a working guide, but the shape of the population is the real driver. Chapter 07 · Sums of Random Variables and the Central Limit Theorem
- Why do we take the logarithm of the likelihood?
- The logarithm turns a product of many terms into a sum, which is far easier to differentiate, and because the logarithm is increasing it does not move the location of the maximum. Chapter 08 · Point Estimation with Likelihood and Moments
- When should I use t instead of z for a mean?
- Use t whenever the population standard deviation is estimated from the same sample, which is the usual case. Use z only when σ is genuinely known from outside the data. Chapter 09 · Confidence Intervals and Sample Size
- Does failing to reject the null hypothesis prove it is true?
- No. It means the data are compatible with the null hypothesis at the chosen level. Like a not guilty verdict, it reflects insufficient evidence rather than proof of innocence. Chapter 10 · Hypothesis Tests for Means, Proportions and Variances
- What is the difference between a test of independence and a test of homogeneity?
- Independence uses one sample classified by two variables; homogeneity uses separate samples from several populations. The expected counts and degrees of freedom are computed the same way. Chapter 11 · Power and Chi-Square Tests
How to study for the exam
Work in three passes per content block. First, rebuild each chapter's key results from their definitions so you know why each formula holds. Second, sort mixed word problems by model or test before solving any of them, because recognition is where exam marks are won or lost.
Third, time yourself on complete solutions written the way the homework rubric rewards: correct, complete and explained, with three significant digits at the end.
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