STAT 400 Chap.4 Binomial, Geometric, Poisson and Related Models
Binomial, Geometric, Poisson and Related Models
Recognition is the main skill
The second content block of the course is built around a family of named discrete distributions. Each one comes from a specific story, and once the story is identified the pmf, mean and variance follow from a known formula.
The first step of any question is therefore to decide what X counts and under what conditions, rather than to start computing.
Counting successes in a fixed number of trials
The binomial model counts successes in n independent trials that share the same success probability p. Its mean np and variance np(1 − p) are worth knowing on sight.
When a question asks for at least a certain number of successes, the complement usually saves time, since it replaces a long sum with a short one.
The Bernoulli distribution is the one-trial case, and the binomial is a sum of independent Bernoulli variables, a fact the central limit theorem uses later.
Waiting for successes
If the number of trials is not fixed and you wait for a success, the variable changes family.
The geometric distribution gives the trial on which the first success occurs, with mean 1/p, and its tail P(X > k) = (1 − p)^k produces the memoryless property. The negative binomial generalises this to the trial of the r-th success, with mean r/p.
Note the convention: this course counts trials, not failures, so the support starts at 1 for the geometric and at r for the negative binomial.
Rates, finite batches and several categories
The Poisson model counts events occurring at a constant average rate over an interval of time or space. Its parameter is the mean count for the interval in the question, so rates must be rescaled first.
Its mean and variance are equal, and it approximates a binomial with large n and small p. When items are drawn without replacement from a finite batch, the success probability changes from draw to draw and the hypergeometric model applies.
For trials with more than two outcomes, the multinomial model extends the binomial, and each single category count on its own is binomial.
Checking assumptions
Many exam questions reward a sentence that states the assumptions: independence of trials, constant p, a fixed n, or a constant rate. Naming the assumption also tells you what would break the model if the story were changed.
What this chapter covers
- 01
Bernoulli and binomial distributions
- 02
Geometric waiting times and memorylessness
- 03
Negative binomial distribution
- 04
Poisson rates and rescaling
- 05
Hypergeometric sampling without replacement
- 06
Multinomial counts
Worked example · free
Pick the model, then compute
- 1Model: events at a constant average rate over time, so the count is Poisson.
- 2Rescale: 10 minutes is one sixth of an hour, so λ = 1 and P(X = 0) = e^(−1) ≈ 0.368.
- 2For 25 minutes, λ = 6 × 25/60 = 2.5, which is also the expected count.
Key terms
- Binomial Distribution
- The distribution of the number of successes in n independent trials with a common success probability p.
- Geometric Distribution
- The distribution of the trial number on which the first success occurs in independent trials with success probability p.
- Memoryless Property
- The property that past failures do not change the distribution of the remaining waiting time, P(X > j + k | X > j) = P(X > k).
- Poisson Distribution
- The distribution of the number of events in a fixed interval when events occur independently at a constant average rate λ.
- Hypergeometric Distribution
- The distribution of the number of successes in a sample drawn without replacement from a finite population.
- Multinomial Distribution
- The joint distribution of category counts in n independent trials that each fall into one of several categories.
- Negative Binomial
- The distribution of the trial on which the r-th success occurs, with mean r/p.
Binomial, Geometric, Poisson and Related Models FAQ
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.
When can the Poisson approximate the binomial?
When n is large and p is small, so that np is moderate. Set λ = np; the approximation is then accurate to about the third decimal place for typical textbook values.
Why does the hypergeometric variance have an extra factor?
Sampling without replacement makes successive draws negatively dependent, which reduces variability. The finite population correction (N − n)/(N − 1) captures that reduction and approaches 1 when the batch is large.
Exam move
Make a one-page table of the six models with the story, support, pmf, mean and variance, then sort ten short word problems into the table before solving any of them. Speed at recognition is what the Exam 2 block rewards.
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