Premed · Premed · Statistics Biostatistics

Lecture 9: The Binomial and Poisson Distributions

Statistics / Biostatistics

Learning Objectives

By the end of this lecture, students will be able to:

1. Identify the conditions for a binomial experiment
2. Calculate binomial probabilities, mean, and variance
3. Identify the conditions for a Poisson process
4. Calculate Poisson probabilities, mean, and variance
5. Use the Poisson distribution as an approximation to the binomial
6. Apply both distributions to clinical and epidemiological problems

Lecture Content

I. The Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. Four conditions, remembered by the acronym BINS, must be satisfied. Each trial must be Binary, having exactly two outcomes (success or failure). The trials must be Independent of one another. The Number of trials (n) must be fixed in advance. And the Same probability of success (p) must be constant across all trials.

If X follows a Binomial(n, p) distribution, the probability mass function is P(X = k) = C(n,k) p^k (1-p)^(n-k), for k = 0, 1, 2, ..., n. The expected value is E(X) = np, the variance is Var(X) = np(1-p), and the standard deviation is SD(X) = sqrt(np(1-p)).

II. Binomial Distribution: Worked Examples

Consider a drug with a 70% cure rate. If 10 patients are treated independently, the probability of exactly 8 being cured is P(X = 8) = C(10,8) (0.70)^8 (0.30)^2 = 45 0.05765 0.09 = 0.2335. To find the probability that at least 9 of 10 patients are cured, compute P(X >= 9) = P(X=9) + P(X=10). Here P(X=9) = C(10,9) (0.70)^9 (0.30)^1 = 10 0.04035 0.30 = 0.1211, and P(X=10) = (0.70)^10 = 0.0282, giving P(X >= 9) = 0.1211 + 0.0282 = 0.1493.

The shape of the binomial distribution depends on p. It is symmetric when p = 0.5, right-skewed when p < 0.5, and left-skewed when p > 0.5. As n increases, the distribution approaches a normal distribution, provided np >= 5 and n(1-p) >= 5.

<image>A three-panel figure showing binomial distributions with n = 20 and varying p. Panel A: p = 0.5 (symmetric, bell-shaped). Panel B: p = 0.1 (right-skewed, most probability mass near 0-4). Panel C: p = 0.9 (left-skewed, most probability mass near 16-20). Each panel is a bar chart with x on the horizontal axis and P(X=x) on the vertical axis. The mean (np) is marked with a vertical dashed line on each.</image>

III. Clinical Applications of the Binomial Distribution

The binomial distribution appears throughout clinical research. It models the number of patients responding to treatment out of n treated, the number of positive test results in a batch of n tests, the number of correctly classified diagnoses out of n cases, genetic outcomes such as the probability of k affected offspring out of n children, and quality control scenarios like the number of contaminated samples in a batch.

IV. The Poisson Distribution

The Poisson distribution models the number of events occurring in a fixed interval of time, space, or volume. Its conditions require that events occur independently of each other, at a constant average rate (lambda), and that two events cannot occur at exactly the same instant. The probability of an event in a very small interval must be proportional to the interval length.

If X follows a Poisson(lambda) distribution, the PMF is P(X = k) = (e^(-lambda) * lambda^k) / k!, for k = 0, 1, 2, .... The expected value is E(X) = lambda, and the variance is also Var(X) = lambda. This equality of mean and variance is a key property of the Poisson distribution.

V. Poisson Distribution: Worked Examples

Suppose a hospital emergency department sees an average of 4 cardiac arrests per week. The probability of exactly 2 in a given week is P(X = 2) = (e^(-4) 4^2) / 2! = (0.01832 16) / 2 = 0.1465. The probability of no cardiac arrests in a week is P(X = 0) = e^(-4) = 0.0183.

The Poisson distribution is always right-skewed but becomes more symmetric as lambda increases. For large lambda (>= 20), the Poisson approximates a normal distribution with mu = sigma^2 = lambda.

VI. Poisson Approximation to the Binomial

When n is large and p is small, the binomial distribution can be approximated by the Poisson distribution. The rule of thumb is n >= 20 and p <= 0.05 (or np <= 10), with lambda set to np. This simplifies calculations when n is very large.

For example, in a population of 10,000 people, suppose the probability of a rare genetic mutation is 0.0003. The probability of exactly 2 people having the mutation is approximated using lambda = np = 10000 0.0003 = 3, giving P(X = 2) = (e^(-3) 3^2) / 2! = (0.0498 * 9) / 2 = 0.2240.

<image>An overlay comparison of binomial and Poisson distributions. Panel A: Binomial(n=100, p=0.03) displayed as bars alongside Poisson(lambda=3) displayed as dots connected by a line, showing close agreement. Panel B: Binomial(n=20, p=0.3) versus Poisson(lambda=6), showing the approximation is less accurate when p is not small. Each panel includes a legend and annotation of the conditions for valid approximation.</image>

VII. Comparison of Binomial and Poisson

VIII. Overdispersion and Underdispersion

In real data, the variance may not equal the mean as the Poisson distribution assumes. Overdispersion occurs when the variance exceeds the mean, and it is common in biological data due to clustering or heterogeneity among subjects. When overdispersion is present, the negative binomial distribution may be a more appropriate model. Underdispersion, where the variance is less than the mean, is less common. Checking whether the variance approximately equals the mean is an important step in validating the Poisson assumption before applying Poisson-based methods.

<image>A bar chart comparing the observed frequency distribution of rare adverse events in a clinical trial (bars) with the expected Poisson distribution (dots with connecting line) using the same mean. Panel A: Good fit -- observed frequencies closely match Poisson expectations. Panel B: Poor fit with overdispersion -- observed data show more zeros and more high counts than Poisson predicts, suggesting a negative binomial model may be more appropriate. Chi-square goodness-of-fit test results are annotated on each panel.</image>

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