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3.4 Theoretical Distributions 43

An example is a rolling die with N=6 faces. A discrete variable such as the
faces of a die can only take a countable number of values x. The probability
of each face is 1/6. The probability density function of this distribution is

The corresponding cumulative distribution function is

where x takes only discrete values, x=1, 2, …, 6.

Binomial or Bernoulli Distribution

A binomial or Bernoulli distribution, named after the Swiss scientist James
Bernoulli (1654-1705), gives the discrete probability of x successes out of
N trials, with probability p of success in any given trial (Fig. 3.5). The prob-
ability density function of a binomial distribution is

Probability Density Probability Density
Function p=0.1 Function p=0.3
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
f(x) 0.4 f(x) 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
x x
a b

Fig. 3.5 Probability density function f(x) of a binomial distribution, which gives the
probability p of x successes out of N=6 trials, with probability a p=0.1 and b p=0.3 of
success in any given trial.