Binomial distribution, Mathematics

Binomial Distribution

Consider a batch of N light bulbs. Each bulb may be defective (S) or non-defective (F). The experiment involves selecting a light bulb and checking whether it is S or F. This experiment is called a Bernoulli Experiment since it has only two outcomes Success and Failure. Suppose it is known that there are M defective light bulbs in the batch. If we represent success by 1 and failure by 0, then

P (Success) = P (X = 1)           = M/N = p (say)

P (Failure)  = P (X = 0)             = 1 - p = q (say)

X is said to be a random variable with Bernoulli distribution.

(Notice that a Bernoulli experiment can always be replicated by a (biased) coin with Head = 1, Tail = 0, P(1) = p)

Suppose the Bernoulli experiment is repeated n times under the same condition. That is, after the light bulb is tested, it is put back into the batch. This way, the probabilities p and q remain unchanged. (This type of sampling is called Sampling with Replacement.)

Let X = Number of successes in n trials.

Then, P(X = x) =   1049_binomial distribution.png px qn - x, x = 0, 1, 2, ..., n  where   1089_binomial distribution1.png

We sum up the Bernoulli Process as follows:

1. Each trial has only two possible outcomes.

In our example, the two possible outcomes are whether a bulb is defective or non-defective.

2. The probability of the outcome of any trial remains fixed over time.

In our example, the probability of the bulb being defective or non-defective remains fixed throughout.

3. The trials are statistically independent.

In our example, the outcome of the bulb being defective or non-defective does not affect the outcome of any other bulb being so.



Find the probability of getting exactly three heads in 4 tosses of a biased coin, where

P(H) = 3/4 and P(T) = 1/4

P(X = 3)= 

2322_binomial distribution2.png (0.75)3 (0.25) = 4 x (0.75)3 x (0.25)



It can be shown for the Binomial Distribution

m = E(x)  = np

s2 = V(X) = npq

Posted Date: 9/15/2012 1:34:28 AM | Location : United States

Related Discussions:- Binomial distribution, Assignment Help, Ask Question on Binomial distribution, Get Answer, Expert's Help, Binomial distribution Discussions

Write discussion on Binomial distribution
Your posts are moderated
Related Questions
The diagrams drawn to given sets are called as Venn diagrams or Eule -Venn diagrams. Here given the universal set U by points within rectangle and the subset A of the set U given b

construct of tangents a circle from an external point when its centre is not known

Calculate the Kendaul''s correlation cofficient for a given data.

I have about 6 Statistics questions, can anyone help me?


The sum of the square of a number and 12 times the number is -27. What is the smaller probable value of this number? Let x = the number.  The statement that is "The sum of the

?[1,99] x^5+2x^4+x^3+5x^2+6x+2÷x^2+2x

The division algorithm says that when a is divided by b, a unique quotient and remainder is obtained. For a fixed integer b where b ≥ 2, consider the function f : Z → Z given by f(

Derivatives of Hyperbolic Functions : The last set of functions which we're going to be looking at is the hyperbolic functions.  In several physical situations combinations of e

1) Find the maxima and minima of f(x,y,z) = 2x + y -3z subject to the constraint 2x^2+y^2+2z^2=1 2) Compute the work done by the force ?eld F(x,y,z) = x^2I + y j +y k in moving