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.

Example

 

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)

=

0.421875  

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
Evaluate following sin 2 ?/3   and sin (-2 ?/3) Solution: The first evaluation in this part uses the angle 2 ?/3.  It is not on our unit circle above, though notice that  2 ?/

if triangle abc is similar to def and ab/de=3/4 find the ratio af their perimeter and area

Solution : We'll require the first and second derivative to do that. y'(x) = -3/2x -5/2                                     y''(x) = 15/4x -7/2 Plug these and also the funct

how to compare fractions

#question what is input and output analysis


Suppose we are required to find the difference between 3abc and 7abc. We look at two scenarios. The value we would obtain by subtracting a larger quantity from th

Example:  find out the slope of equations and sketch the graph of the line.                         2 y - 6x = -2 Solution To get the slope we'll first put this in slope

Derivative for the trig function: We'll begin with finding the derivative of the sine function. To do this we will have to utilize the definition of the derivative. It's been wher