Bernoulli & Binomial
Distributions
Trial
A single performance to get some information is called trial.
Bernoulli Trial
A trial which has two possible outcomes i.e. success and failure is called Bernoulli trial.
Bernoulli Experiment
A Bernoulli experiment is a random experiment, the outcome of which can be classified
in one of two mutually exclusive and exhaustive ways (success or failure). A sequence of
Bernoulli trials occurs when a Bernoulli experiment is repeated independently, so that the
probability of success remains the same from trial to trial.
Bernoulli Distribution
In Bernoulli trials the two outcomes, success or failure, are denoted by 1 & 0. The
probability function of X is given by
q if x = 0
f(x) = p if x = 1
0 otherwise
(OR)
If X ~ b(x; 1, p) then
x 1 x
P X = x = p q
; x = 0, 1.
Where
X → Random variable.
x → Value of random variable (Number of successes).
p → Probability of success.
q → Probability of failure.
p is parameter of the Bernoulli distribution.
p + q = 1 q = 1 – p
Mean and Variance of the Bernoulli Distribution
Mean = p,
Variance = pq,
S.D. = pq
In Bernoulli distribution, Mean > Variance
Binomial Experiment
An experiment having two possible outcomes repeated independently under the same
conditions for fixed value of probability of success for a fixed number of time “n” is
called “Binomial experiment”.
Binomial Random Variable
The random variable X which represents the number of successes in a binomial
experiment is called a binomial random variable. The binomial variable is a discrete
variable which can assume any of the values x = 0, 1, 2, . . . , n.
