Consider a random experiment which has only two outcomes that can be classified
as "success" and "failure". We call such an experiment a Bernoulli experiment (after the Swiss mathematician James Bernoulli).

Let X = 0 when the experiment is a failure, and X = 1
when it is a success.
The pdf of X is given by P(X = 1) = p and P(X
= 0) = 1 - p where 0 < p <
1.
The rv X is said to be a Bernoulli rv or X follows a
Bernoulli distribution.

inomial Distribution

Suppose a Bernoulli experiment is repeated n times independently.
Let Y = number of successes that occur in the n trials.
The Y is said to follows a binomial distribution
with parameter (n, p), written as Y
~ B(n, p).

Thus a Bernoulli distribution is just a binomial distribution with parameters
(1, p).

It is clear that Y takes the values 0, 1, ..., n.

The pdf of Y is given by P(Y
= y) = ^{n}C_{r}p^{r}q^{n}^{-r},
where q = 1 - p, for y
= 0, 1, ..., n.

onditions For Binomial Distribution

A fixed number of trials

Only 2 outcomes in each trial

P(success) is the same for every trial

Trials are independent

xpectation & Variance

If X ~ B(n, p), then E(X)
= np, Var(X) = npq.

ecurrence Formula

P(X = x + 1)

¾¾¾¾¾¾

P(X = x)

=

(n - x)p

¾¾¾¾

(x + 1)q

itting A Binomial Distribution

Example: Fit a binomial distribution to the following data:

x

0

1

2

3

4

5

Total

f

6

16

21

12

5

2

62

Solution:

Take n = 5 and np = Sfx/Sf
= 119/60 = 124/62 = 2.
Therefore p = 0.4.

Let X ~ B(5, 0.4).
P(X = 0) = 0.6^{5} = 0.0778.
Using the recurrence formula,