11 Jun 1999

ernoulli Distribution

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) = nCr prqn-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) = 0.65 = 0.0778.
Using the recurrence formula,

P(X = 1) = 0.2592
P(X = 2) = 0.3456
P(X = 3) = 0.2304
P(X = 4) = 0.0768
P(X = 5) = 0.0102

Multiply by 60 to give the expected frequency.

 x 0 1 2 3 4 5 Total fE 5 16 21 14 5 1 62

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