BIOL2001 Chap.7 Binomial, Poisson and Normal Distributions
Binomial, Poisson and Normal Distributions
Three models for three kinds of variable
A probability distribution assigns a probability to every possible outcome of a random variable. This chapter separates discrete variables, which take countable values, from continuous ones, which can take any value in an interval, and matches each to a model.
The binomial describes the number of successes in a fixed number of independent two-outcome trials; the Poisson describes the number of independent events in an interval; the normal describes a continuous measurement clustered symmetrically around its mean.
Formulas, counting and areas
A computed bar chart shows the binomial distribution for six seeds that each germinate with probability 0.6. Counting rules for permutations and combinations explain the n choose k term in the binomial formula.
A worked Poisson example rescales a rate to the length of a transect, and a computed normal curve illustrates the empirical rule, with probability as area under the curve and zero probability for any single exact value.
What this chapter covers
- 01
Discrete and continuous random variables
- 02
Bernoulli trials and the binomial formula
- 03
Mean np and standard deviation of a binomial
- 04
Permutations, combinations and n choose k
- 05
Poisson rates rescaled to the interval
- 06
Normal density and the empirical rule
- 07
R functions such as pbinom and pnorm
Worked example · free
Germinating seeds with the binomial formula
- 1Count the arrangements: C(6, 4) = 15 ways for four of six seeds to germinate.
- 1Multiply by the probability of one arrangement: 0.64 × 0.42 = 0.1296 × 0.16 = 0.020736.
- 1P(X = 4) = 15 × 0.020736 = 0.311.
- 1Mean np = 6 × 0.6 = 3.6 seeds; standard deviation √(6 × 0.6 × 0.4) = √1.44 = 1.2 seeds.
Key terms
- Bernoulli Trial
- A random experiment with exactly two outcomes, called success and failure.
- Binomial Coefficient
- The number of ways to choose k items from n when order does not matter, written C(n, k).
- Poisson Rate
- The expected number of events per unit of time, area, distance or volume.
- Empirical Rule
- The pattern that about 68%, 95% and 99.7% of normal values lie within one, two and three standard deviations of the mean.
- Continuous Variable
- A variable that can take any value in an interval, such as body mass or length.
- Permutation
- An arrangement of items in which the order matters.
Binomial, Poisson and Normal Distributions FAQ
How do I know whether to use the binomial or the Poisson distribution?
Use the binomial when there is a fixed number of trials, each with two outcomes and the same probability of success. Use the Poisson when you count independent events in an interval of time, area or distance with no fixed upper limit on the count.
Why is the probability of an exact value zero for a normal variable?
Probabilities for continuous variables are areas under the density curve. A single exact value is a point with no width, so the area above it is zero. Questions about normal variables therefore always ask for ranges, such as the probability of a value above 45 mm.
What does it mean that a Poisson mean equals its variance?
For a Poisson variable the variance equals the mean, so its standard deviation is the square root of the mean. A count with mean 5 has standard deviation about 2.24. When real counts vary much more than this, the Poisson model is overdispersed, a problem met again in Part III.
Assessment move
Make a one-page sheet with each distribution's conditions, parameters, mean and standard deviation, and use it to label ten practice scenarios with the right model. Compute two binomial and two Poisson probabilities by hand, then check them in R. Sketch a normal curve for every area question and shade the region before computing.
Before each calculation, write one line confirming the model's conditions in the context of the question, such as independent seeds or a constant rate along the transect; that line is evidence of understanding that a bare number cannot show.
Working through Binomial, Poisson and Normal Distributions in BIOL2001? Sia is AskSia’s AI Statistics tutor — ask any BIOL2001 Binomial, Poisson and Normal Distributions question and get a clear, step-by-step explanation grounded in how BIOL2001 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
STAT1008 Quantitative Research Methods · BIOL1030 Diversity of Life · STAT7038 Regression Modelling
