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.,
1.0 Finite Population (N), D(D<N),
Number that fall into D = (x), Random Sample (n) ,
Use Hypergeometric Distribution
2.0 Infinitely Large Population,
P = fraction defective, x = number of Nonconforming items in population, n =
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
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
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
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
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
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
The trials are independent.
6). A finite population
A population consisting of a fixed
number of known individuals, objects, or measurements.
7). The Hypergeometric Distribution
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
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 (a criteria
for the binomial distribution is
that the probability of success
remains the same from trial to
the size of the
sample n is greater than 5% of the
size of the population N .
The hypergeometric distribution has the following
There are only 2 possible outcomes.
The probability of a success is not the same on each
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
of the event
to the size
is used to
of errors in
is formed by
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
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