Counts and waiting times
Follow one process from a single success to counts, intervals, and accumulated waiting.
- Step 1
Read a distribution
Distinguish point probabilities from areas and cumulative probabilities.
Checkpoint: Does density 0.4 mean probability 40% at that point?
No. For a continuous model, probability is area over an interval. A point has zero probability even where density is positive.
- Step 2
One trial: Bernoulli
Represent a yes/no outcome as zero or one.
Checkpoint: Does one Bernoulli draw estimate p accurately?
One draw is just zero or one. Its distribution has mean p, but estimating an unknown p requires repeated observations and an uncertainty assessment.
- Step 3
Many trials: binomial
Count successes in a fixed number of independent opportunities.
Checkpoint: What breaks if success probability changes across trials?
The count is generally no longer binomial with one common p. Independence and an identical success probability are separate assumptions.
- Step 4
Events in a window: Poisson
Replace trial counts with a rate and an observation window.
Checkpoint: At two arrivals per minute, what is the Poisson mean for five minutes?
Ten arrivals. The parameter is expected count for a specified window: rate multiplied by duration.
- Step 5
Wait for one event
See the same events through the gaps between arrivals.
Checkpoint: What happens to the expected wait if the constant rate doubles?
It halves, since the exponential mean is one divided by the rate.
- Step 6
Wait for several events
Add independent exponential waiting times.
Checkpoint: When does a gamma variable describe time to the third arrival?
For a homogeneous Poisson process, the sum of three independent exponential gaps has gamma shape 3 with the same rate.
After each lesson, try to explain its main result without looking at the formula. Then change one assumption: which part of the answer would change?