Binomial distribution
A binomial distribution is the most commonly encountered distribution that arises from a sequence of independent Bernoulli trials.
Definition
If the following conditions hold:
then the total number of successes, \(X\), has a binomial distribution with parameters n and \(\pi\).
\[ X \;\; \sim \;\; \BinomDistn(n, \pi) \]In most practical applications, the parameter \(\pi\) is an unknown constant, but occasionally we know its value.
Number of sixes when four dice are rolled
The number of sixes, \(X\), therefore has a binomial distribution,
\[ X \;\; \sim \; \; \BinomDistn(n=4, \pi=\frac 1 6) \]Sex of reptiles
In many reptiles, sex is partly determined by the incubation temperature of the eggs. In an experiment with 10 lizard eggs incubated at 25ºC, the number of males hatching, \(X\), has a binomial distribution since
Although we can often argue from the context that the assumptions behind the binomial distribution hold, there may be some doubt over this.
Rainy days in week
The table below shows the number of rainy days each week during a year at Balcombe, Sussex in the UK.
Number of days in week with measurable rain, x |
Number of weeks in year (frequency) |
---|---|
0 1 2 3 4 5 6 7 |
4 9 15 11 6 4 2 1 |
52 |
We first consider whether the random variable X — the number of rainy days in a single week — might have a binomial distribution.
Although the assumptions underlying the binomial distribution are unlikely to hold exactly, with short weather cycles the effect of dependence of adjacent days' weather may be slight. As an approximation, we can therefore tentatively make the assumption of independence, and model the number of rainy days in a week with a binomial distribution.
\[ X \;\; \sim \; \; \BinomDistn(n=7, \pi) \]A further complication with the data set shown above is that the 52 different weeks are at different times in the year. Are the 52 binomial distributions underlying the 52 counts all the same? As rainfall is not strongly seasonal in the south of England, the value of \(\pi\) is likely to be at least approximately the same in each week, so the 52 values in the table are likely to be at least approximately a random sample from the same distribution,
\[ X_i \;\; \sim \; \; \BinomDistn(n=7, \pi) \quad \quad \text{for }i=1, ..., 52\]We should however remember our doubt about the assumption of independence and examine the data later for evidence that this assumption does not hold.