BINOMIAL DISTRIBUTION

GROUP MEMBERS

• ALEENA AKHTAR(11-EL-23)• ANOOSH KHAN(11-EL-39)• ZARISH QAISER(11-EL-19)• TAHIRA MAHAM FATIMA(11-EL-29)• MUQADAS RIDA FATIMA(11-EL-27)

ALEENA AKHTAR (11-EL-23)

DISTRIBUTION

STATISTICS AND ENGINEERING• The field of statistics deals with the

collection, presentation, make decisions, solve problems, and design products.

• Engineering statistics combines engineering and statistics

• Statistical approaches can provide the basis for making decisions.

• Reliability engineering use statistics.• Quality control and process control use

statistics.• Time and methods engineering use

statistics.

DISTRIBUTION

The probability that the variable takes a value less than or equal to specified x.

Arrangement of values of a variable showing their observed or theoretical frequency of occurrence

The distribution is of discrete and continuous variable.

Description of the relative numbers of times each possible outcome will occur.

It is non decreasing.

It is non negative.

Distributions have different shapes.

EXAMPLES• Statistical distribution is used in many field of real life.• An engineer must determine the strength of supports for

generators at a power plant. A number of those available must be loaded to failure and their strengths will provide the basis for assessing the strength of other supports.

• Groups of networked computers which have same goal for work.

• Bits are sent over a communications channel in packets of 12. If• the probability of a bit being corrupted over this channel is 0.1

and such errors are independent, what is the probability that no more than 2 bits in a packet are corrupted?

• Used in processing language to assign distribution.

ANOOSH KHAN (11-EL-39)

BINOMIAL DISTRIBUTION

TYPES OF DISTRIBUTION• Many probability distributions are so

important in theory or applications that they have been given specific names.

• The main types of distribution are as follows.

binomial distribution Hyper geometric distribution

Poisson distribution Negative binomial distribution

Geometric distribution Multinomial distribution

• The types are related to each other in some way.

HISTORY OF BINOMIAL• Swiss mathematician Jakob Bernoulli, in a

proof published posthumously in 1713, determined that the probability .Hence the name binomial distribution.

• In 1936 the British statistician Ronald Fisher used the binomial distribution to publish evidence of possible scientific theory.

• The work of jakob was published in ARTS CONJECTANDI IN BASEL IN 1713.

• The binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent each of which yields success with probability p

BINOMIAL DISTRIBUTION• To model the number of successes in a sample of size n drawn

with replacement from a population of size N.• Without replacement, it is called as the hyper geometric

distribution.

• The parameters are n,p and denoted by (x;n,p).• The important points for this are as follows:

Two out comes: success and failureThe successive trails are independent.

The probability of success remain constantThe experiment is repeated at fixed no of times.

ZARISH QAISER (11-EL-19)

APPLICATIONS & EXAMPLES

GENERAL APPLICATIONS• Binomial distribution occurs whenever

there is a success and fail.• The outcome may be trail or head,

success and failure, wrong and right,• Some general examples are:

A fair coin is tossed five times. probability of obtaining the head?

A baseball player comes to bat 4 times in a game. The chance of a strike-out for this

player is 30%.

APPLICATIONS IN ENGINEERING• A motor Machine produces 20% defective components. In a

random sample of 6 components. Determine the probability that: There will be 3 defective components

• The output of an automated machine is inspected by taking samples of 6 items. If the probability of a defective item is 0.25, find the probability of having ,two defective items.

• The probability of obtaining a defective resistor is given by 1/10 .In a random sample of 9 resistors, what is the probability of 3 defective resistors.

• The probability of successful transmission of signal to receiver is 0.25. if there are 6 signals what is the probability of successful reception of 3 signals

APPLICATIONS IN REAL LIFE.• If a new drug is introduced to cure a disease

then it either cure the disease or doesn’t.• If a person purchase a lottery ticket then he

is either going to win it or not.• the number of successful sale calls.• The no of defective products in a

production run.• The experience of a house agent indicated

that he can provide suitable accommodation for 75 percent clients. if 6 clients approach to him then what is the probability for 4 clients?

TAHIRA MAHAM FATIMA (11-EL-29)

PROPERTIES OF DISTRIBUTION

MAIN PROPERTIES• A binomial experiment (also known as a

Bernoulli trial) is a statistical experiment that has the following properties:

The experiment consists of n repeated trials. Each trial can result in just two possible

outcomes. We call one of these outcomes a success and the other, a failure.

The probability of success, denoted by P, is the same on every trial.

The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

PROPERTIES OF DISTRIBUTION

• The mean (also know as average), is obtained by dividing the sum of observed values by the number of observations,

In binomial distribution the mean is µ=np• The standard deviation gives an idea of how close the entire set

of data is to the average value.

In binomial the variance is given as: • The moments of distribution about origin can be calculated.

EXAMPLES• The probability of obtaining a defective resistor is given by

1/10 In a random sample of 9 resistors what is the mean number of defective resistors you would expect and what is the standard deviation?

Mean = n x p = 9 x 0.1 = 0.9 SD = √(9x0.1x0.9) = √0.81 = 0.9

• A motor Machine produces 20% defective components. In a random sample of 6 components. What is the mean of this distribution.

Mean = n*p = 6*0.2=1.2SD==0.97

MUQADAS RIDA FATIMA (11-EL-27)

FORMULAS AND OTHER DISTRIBUTIONS

FORMULAS• A recurrence relation is an equation that recursively defines a

sequence each further term of the sequence is defined as a function of the preceding terms.

• If distribution is multiplied by N, then the resulting distribution

is binomial frequency.

N• Moment generating function is

Mo(t)==

RELATIONS WITH OTHER DISTRIBUTIONS

Hyper geometric is the distribution like binomial one but with replacement of the elements.

If N is large then it becomes binomial distribution.

The Poisson distribution is for rare events .the Poisson distribution cases can be handled with binomial one.

The negative distribution is when the trials are variables to reach the fixed success.

The geometric distribution is the one in which the trials are variable to reach the first success.

The geometric distribution is the special case of negative distribution

CONCLUSION

Statistics play a very important role in engineering. This is because most engineering projects have to be

precisely calculated with every little detail recorded in order for it to work and fit together correctly