Uniform and exponential distributions
Beyond the binomial and normal, a few other distributions model waiting times and equally likely outcomes, and each has a clear story behind it.
Why learn named distributions?
Named distributions are templates for common random situations. If you recognise the story, such as 'count of successes', 'waiting time', or 'equally likely values', you instantly know the formulas for probabilities, the mean, and the spread. Choosing the right template is a core modelling skill.
The discrete uniform distribution
When each of \(k\) values is equally likely, as with a fair die, each has probability \(\frac{1}{k}\). For a die, the mean is \(3.5\). Random number generators and fair lotteries are built to produce uniform outcomes.
The continuous uniform distribution
A continuous uniform variable on \([a, b]\) has a flat density of height \(\frac{1}{b-a}\). Probabilities are rectangle areas: if a bus arrives uniformly within a 20-minute window, \(P(\text{wait} \le 5) = \frac{5}{20} = 0.25\). The mean is the midpoint \(\frac{a+b}{2}\) and the variance is \(\frac{(b-a)^2}{12}\).
f(x) = \frac{1}{b-a},\ a \le x \le bThe exponential distribution
Waiting times between random events, such as phone calls or radioactive decays, often follow an exponential distribution with rate \(\lambda\): \(P(X > t) = e^{-\lambda t}\), mean \(\frac{1}{\lambda}\). If a help desk receives 6 calls per hour, the mean gap is 10 minutes, and the chance of waiting more than 20 minutes is \(e^{-2} \approx 0.135\).
P(X > t) = e^{-\lambda t}Memorylessness
The exponential distribution is memoryless: having already waited 10 minutes does not change the distribution of the remaining wait. This fits processes with no 'wear', like radioactive decay, but not lightbulbs that wear out with age. It pairs with the Poisson distribution: if event counts are Poisson, gaps between events are exponential.
The t distribution
When estimating a mean from a small sample with unknown population standard deviation, the standardised mean follows a t distribution. It looks like the normal but with heavier tails, reflecting extra uncertainty. It has a parameter called degrees of freedom (\(n - 1\) for one sample); as this grows, the t distribution approaches the standard normal.
The chi-square distribution
The chi-square distribution arises from sums of squared standard normal values. It is skewed right, takes only positive values, and is used to test whether observed category counts match expected counts, and whether two categorical variables are associated.
Choosing a distribution
Count of successes in fixed trials: binomial. Trials until first success: geometric. Count of events in an interval: Poisson. Waiting time between events: exponential. Equally likely values: uniform. Measurements that cluster symmetrically: normal. Small-sample means: t. Category counts: chi-square.
Key takeaways
Uniform distributions have flat probabilities; exponential distributions model memoryless waiting times with mean \(1/\lambda\). The t distribution handles small-sample means, and chi-square handles categorical counts. Match the story of the problem to the distribution.
Flashcard
Each of k values has probability 1/k
Check your understanding
Calls arrive at an average rate of 4 per hour. What is the mean waiting time between calls?