CVEN9407 Chap.2 Probability Distributions for Transport Data
Probability Distributions for Transport Data
Every test and model later in the course rests on a handful of distributions, and this chapter assembles them. It starts with events, probability rules and random variables, then measures centre and spread, including why the sample variance divides by n minus one, and the higher moments that the Jarque-Bera normality test uses.
It works a normal-area problem on travel times step by step, links the chi-square, t and F distributions, and introduces the Bernoulli, binomial and Poisson distributions that describe crash counts.
The ideas here return throughout the term.
Skewness and the Jarque-Bera test reappear when you check regression residuals, the t and F distributions carry every coefficient test, and the Poisson assumption that the variance equals the mean is exactly what the count-model chapter puts to the test. Learning the distributions once, carefully, makes those later chapters much shorter.
What this chapter covers
- 01
Sample spaces, events and the rules of probability
- 02
Discrete and continuous random variables
- 03
Mean, median, mode, variance and standard deviation
- 04
Skewness, kurtosis and the Jarque-Bera test
- 05
Normal areas and the central limit theorem
- 06
Chi-square, t and F distributions
- 07
Bernoulli, binomial and Poisson counts
Worked example · free
Probability of at least two crashes in a month
- 1No crashes: P(0) = e to the power -1.5 = 0.2231.
- 1Exactly one: P(1) = 1.5 x 0.2231 = 0.3347.
- 1At least two is the complement of zero or one: 1 - (0.2231 + 0.3347) = 1 - 0.5578 = 0.4422.
Key terms
- Random variable
- A random variable is a quantity whose value is set by the outcome of a chance experiment, such as next month's crash count.
- Standard deviation
- The standard deviation is the square root of the variance and measures spread in the units of the data.
- Kurtosis
- Kurtosis is a fourth-moment measure of how tall or flat a distribution is, equal to 3 for the normal.
- Central limit theorem
- The central limit theorem states that a sample mean is approximately normal for large samples, whatever the shape of the parent distribution.
- Poisson distribution
- The Poisson distribution describes counts of rare events in a fixed window, with mean and variance both equal to lambda.
- Binomial distribution
- The binomial distribution counts successes in a fixed number of independent trials that share one success probability.
Probability Distributions for Transport Data FAQ
Why does the sample variance divide by n minus one?
Dividing by n would underestimate the population variance on average, because deviations are measured from the sample mean rather than the true mean. Dividing by n minus one removes that bias, which is why software reports it as the sample variance.
Which distribution should I use for crash counts?
Start from the Poisson, the textbook model for counts of rare events, and then check whether the variance is close to the mean. Crash data are often more spread out than a Poisson allows, which leads to the negative binomial model later in the course.
How are the t, chi-square and F distributions related?
All three are built from standard normal variables. A chi-square adds squared normals, a t divides a normal by the root of a scaled chi-square, and an F is a ratio of two scaled chi-squares. A squared t with k degrees of freedom is an F with 1 and k.
Why do crash counts often look skewed to the right?
Most sites or months record few crashes, while a small number record many, so the distribution has a long right tail. In such data the mean sits above the median, and a model that assumes symmetric normal errors fits poorly, which motivates count models.
What does the empirical rule tell me about a normal variable?
For a bell-shaped distribution, roughly 68% of values lie within one standard deviation of the mean and roughly 95% within two. It is a quick way to judge whether a single observation, such as one month's crash total, is unusual.
Assessment move
Make a one-page table of the distributions with what each models, its mean and its variance, and keep it beside you for the rest of the course. Practise normal areas until you sketch and shade the curve without thinking, because the same habit prevents sign slips in every later test. Then compute a few Poisson and binomial probabilities by hand so the count models feel familiar.
When a quiz question gives a mean and a standard deviation, write the standardised value first; most normal-probability errors come from skipping that line.
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