The Bernoulli Distribution
When we first started looking at statistical inference a few weeks ago the binomial distribution was introduced, which reflects the behavior of a coin that has been flipped many times. The Bernoulli distribution is simpler case which the binomial is a generalization of, it represent a coin flipped a single time.
Here’s an example for a biased coin:
bern <- data.frame(Value=c(0,1),Probability=c(.3,.7)) plot(bern,type='h',lwd=5,ylim=c(0,1),main="Bernoulli(0.3,0.7)") points(bern,cex=2,pch=16) # Add large, solid color points
Obviously you can’t get a lot of mileage out of this distribution. It is, however, the source of a lot of common terminology in statistics. By tradition Heads = 1 and Tails = 0 (which part of the reason that in R and many other languages the logical values TRUE and FALSE equal 1 and 0, respectively). The probability of Heads is written p while the probability of tails is written q. Since a coin only has two sides it is always true that q = 1-p and p = 1-q.
Anything that has discrete “success” and “failure” modes can be considered a Bernoulli trial. Flipping a coin is a Bernoulli trial. Rolling a die is a Bernoulli trial if you define every face as either success of failure (ie “a 6 is a success” or “all odds are successes”). Some things are close enough to Bernoulli trial that they can generally be thought of in these terms, for example, many text books use the sex of a newborn as a an example of a Bernoulli trial even though about 1% of people are born with ambiguous or intersex characteristics.
Bernoulli’s name is also given to the concept of a Bernoulli process which is multiple, independent, identical Bernoulli trials. This means that this is more than one trial, trials have no effect on each other, and the probability of success is the same each time. This is what creates a binomial distribution (flipping a coin many times). On the other hand drawing black and white marbles from a box isn’t a Bernoulli process because once you take a black marble out the probability of drawing a black marble decreases, the trials are not identical.
Intuitor 