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Discrete Probability Distributions

 

Determine the Probability
    

    Discrete Distribution, Two Choices (pass or fail)/(accept or reject) Some number fall into class of interest (i.e., nonconforming)
       

1.0  Finite Population (N), D(D<N), Number that fall into D = (x), Random Sample (n) ,
           

Use Hypergeometric Distribution

  • If too Large for Hypergeometric (>400) use Binomial Approximation (p=D/N)

2.0  Infinitely Large Population, P = fraction defective, x = number of Nonconforming items in population, n = random sample

Use Binomial Distribution

  • To Model of Number of defects that occur in a unit (unit area, volume, time, etc. use Poisson Distribution

  • If Sequence of Independent Trials, each with Success = p, x = trial on which rth success occurs use Pascal Distribution


1).  Random Variables 

  • A random variable is a numerical value determined by the outcome of an experiment.

  • A probability distribution is the listing of all possible outcomes of an experiment and the corresponding probability.


2).  Probability Distributions

A probability distribution is the listing of all possible outcomes of an experiment and the corresponding probability.  Probability distributions may be discrete or continuous.

  • Discrete probability distributions - can assume only certain outcomes.  The outcomes are mutually exclusive.  Examples from our book of a discrete distribution are: The number of students in a class. The number of children in a family. The number of cars entering a carwash in a hour. Number of home mortgages approved by Coastal Federal Bank last week.  The sum of the probabilities of the various outcomes is 1.00. The probability of a particular outcome is between 0 and 1.00 and may look the same as continuous distribution.

  • Continuous probability distributions can assume an infinite number of values within a given range.  Examples from the book of a continuous distribution include: The distance students travel to class. The time it takes an executive to drive to work. The length of an afternoon nap. The length of time of a particular phone call.


3).  The mean

  • reports the central location of the data. 

  • is the long-run average value of the random variable. 

  • is also referred to as its expected value, E(X), in a probability distribution. 

  • is a weighted average.

  • where mu represents the mean and P(x) is the probability of the various outcomes x.


4).  The variance

  • measures the amount of spread (variation) of a distribution. 

  • The variance of a discrete distribution is denoted by the Greek letter (sigma squared). The standard deviation is the square root of sigma squared (the variance).


5).  The binomial distribution 

Binomial Distribution: The binomial distribution is a discrete probability distribution that is used when there are only two possible outcomes on a particular trial of an experiment. Another characteristic of the binomial distribution is that the random variable is the result of counts and the probability of success remains the same from one trial to another. The outcome on each trial or experiment is classified as either a success or a failure and the trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial.

Susan Jennings, QNT 554 Summer of 2006   

 

Has the following characteristics:

  • An outcome of an experiment is classified into one of two mutually exclusive categories, such as a success or failure. 

  • The data collected are the results of counts. 

  • The probability of success stays the same for each trial. 

  • The trials are independent.

  • n = the number of trials 

  • x = the number of observed successes 

  • pie = the probability of success on each trial

Mean = 

Variance = 


6).  A finite population

A population consisting of a fixed number of known individuals, objects, or measurements.


7).  The Hypergeometric Distribution

 

Hypergeometric Distribution: One of the criteria for using the binomial distribution is that the probability of success remain the same from trial to trial. If the probability of success does not remain the same then hypergeometric distribution should be applied. Therefore, if the sample is selected from a finite population without replacement and the size of the sample n is mores than 5% of the size of the population N then the hypergeometric distribution is used to determine the probability of a certain number of successes or failures. With hypergeometric distribution the trials are not independent.

Susan Jennings, QNT 554 Summer of 2006   

 

Use the hypergeometric distribution to find the probability of a specified number of successes or failures if:

  1. the sample is selected from a finite population without replacement (a criteria for the binomial distribution is that the probability of success remains the same from trial to trial).

  2. the size of the sample n is greater than 5% of the size of the population N .

The hypergeometric distribution has the following characteristics: 

  • There are only 2 possible outcomes. 

  • The probability of a success is not the same on each trial. 

  • It results from a count of the number of successes in a fixed number of trials.

  • Use the hypergeometric distribution to find the probability of a specified number of successes or failures if: 

    • the sample is selected from a finite population without replacement (recall that a criteria for the binomial distribution is that the probability of success remains the same from trial to trial).

    • the size of the sample n is greater than 5% of the size of the population N .

  • where N is the size of the population, 

  • S is the number of successes in the population, 

  • x is the number of successes in a sample of n observations.


8).  Poisson Probability Distribution

Poisson Distribution:  The Poisson distribution describes the number of times some event occurs during a specified interval.. The probability of the event is proportional to the size of the interval and the intervals do not overlap and are independent. This distribution is used to describe the distribution of errors in data entry or for example, the number of scratches and imperfections in newly painted cars. This distribution is a discrete probability because it is formed by counting.

Susan Jennings, QNT 554 Summer of 2006   

The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller.  The limiting form of the binomial distribution where the probability of success B is small and n is large is called the Poisson probability distribution.

Where (u) is the mean number of successes in a particular interval of time, (e) is the constant 2.71828, and (x) is the number of successes. The mean number of successes : can be determined in binomial situations by n B, where n is the number of trials and B the probability of a success.


 

 

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