Binomial Distribution To understand binomial distributions and binomial probability, it helps to understand binomial experiments and some associated notation; so it is discussed below. below.
Binomial Experimen Ex perimentt A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties: y y
y y
The experiment consists of n of n repeated trials. trials. outco mes a Each trial can result in just two possible out comes. We call one of these outcomes success and the other, ot her, a failure. The probability of success, denoted by P by P , is the same on o n every trial. The trials are independent independent;; that is, the outcome on one o ne trial does not affect the outcome on other trials.
Consider the following statistical experiment. You flip a coin 2 times and count the number of o f times the coin lands on heads. This is a binomial experiment because: y y y y
The experiment consists co nsists of repeated trials. trials. We flip a coin 2 times. Each trial can result in just two possible out comes - heads or tails. The probability of success is constant - 0.5 on every trial. The trials are independent; that t hat is, getting heads on one trial does not affect whether we get heads on other ot her trials. trials.
Notation The following notation is helpful, when we t alk about binomial probability. y y y y y
y
x: x: The number of o f successes that result from the binomial experiment. e xperiment. n: The number of trials in the binomial experiment. P : The probabili pro bability ty of o f success on an individual trial. Q: The probability of failure on an individual trial. (This is equal to 1 - P - P .) .) b( x; x; n, P ): ): Binomial probability - the probability that an n-trial binomial experiment results in exactly x exactly x successes, when the probabili pro bability ty of o f success on an individual trial is P is P . of combinations of n of n things, taken r at r at a time. nCr : The number of combinations
Binomial Distribution A binomial random variable is the number of successes x successes x in n repeated trials of a binomial experiment. The probability The probability distribution of a binomial random variable is called a binomial (also known as a Bernoulli distribution ). distribution (also
Suppose
we flip a coin two times and count the number of heads (successes). The binomial random variable is the number of heads, which can take on values of 0, 1, or 2. The binomial distribution is presented below. Number
of heads Probability
0
0.25
1
0.50
2
0.25
The binomial distribution has the following properties: y y y
The mean of the distribution (x) is equal to n * P . 2 The variance ( x) is n * P * ( 1 - P ). The standard deviation (x) is sqrt[ n * P * ( 1 - P ) ].
Binomial Probability The binomial probability refers to the probability that a binomial experiment results in exactly x successes. For example, in the above table, we see that the binomial probability of getting exactly one head in two coin flips is 0.50. Given x, n, and P , we can compute the binomial probability based on the following formula: Binomial Formula. Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P , then the binomial probability is: x
n-x
b( x; n, P ) = nCx * P * (1 - P)
Example 1 Suppose
a die is tossed 5 times. What is the probability of getting exactly 2 fours?
Solution:
This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is: 2
3
b(2; 5, 0.167) = 5C2 * (0.167) * (0.833) b(2; 5, 0.167) = 0.161
Poisson Distribution Attributes of a Poisson Experiment
A Poisson experiment is a statistical experiment that has the following properties: y y y y
The experiment results in outcomes that can be classified as successes or failures. The average number of successes () that occurs in a specified region is known. The probability that a success will occur is proportional to the size of the region. The probability that a success will occur in an extremely small region is virtually zero.
Note that the specified region cou ld take many forms. For instance, it could be a length, an area, a volume, a period of time, etc.
Notation The following notation is helpful, when we t alk about the Poisson distribution. y
y y y
e: A constant equal to approximately 2.71828. (Actually, e is the base of the natural logarithm system.) : The mean number of successes that occur in a specified region. x: The actual number of successes that occur in a spec ified region. P( x; ): The Poisson probability that exactly x successes occur in a Po isson experiment, when the mean number o f successes is .
Poisson Distribution A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution . Given the mean number o f successes () that occur in a specified region, we can co mpute the Poisson probability based on the following formula: Poisson Formula. Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is . Then, the Poisson probability is: -
x
P( x; ) = (e ) ( ) / x! where x is the actual number of successes that result from the exper iment, and e is approximately equal to 2.71828. The Poisson distribution has the following properties:
y y
The mean of the distribution is equal to . The variance is also equal to .
Example 1
The average number of homes sold by the Acme Realty company is 2 homes per day. What is the probability that exactly 3 ho mes will be sold tomorrow? Solution: y y y
This is a Poisson experiment in which we know the following:
= 2; since 2 homes are sold per day, on average. x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow. e = 2.71828; since e is a constant equal to approximately 2.71828.
We plug these values into the Poisson formula as follows: P( x; ) = (e-) (x) / x! -2 3 P(3; 2) = (2.71828 ) (2 ) / 3! P(3; 2) = (0.13534) (8) / 6 P(3; 2) = 0.180 Thus, the probability of selling 3 ho mes tomorrow is 0.180 . The Normal Distribution (Bell Curve)
In many natural processes, random variation confor ms to a particular probability distribution known as the normal distribution , which is the most commonly observed probability distribution. Mathematicians de Moivre and Laplace used this distribution in the 1700's. In the early 1800's, German mathematician and physicist Karl Gauss used it to analyze astronomical data, and it consequently became known as the Gaussian d ist ri bution among the scientific community. The shape of the normal distribution resembles that of a bell, so it sometimes is referred to as the "bell curve", an example of which follows:
Normal Distribution
The above curve is for a data set having a mean of zero. In general, the normal distribution curve is described by the following probability density function:
Bell
Curv e Charact eristics
The bells curve has the following characteristics: y y y y
Symmetric Unimodal Extends to +/- inf inity Area under the curve = 1
Compl et ely De scribed by Two Paramet ers
The normal distribution can be co mpletely specified by two parameters: y y
mean standard deviation
If the mean and standard deviation are known, then one essentially knows as much as if one had access to every point in the data set. The Empirical Rul e
The empir ical r ul e is a handy quick estimate of the spread of the data given the mean and standard deviation of a data set that follows the normal distribution.
The empirical rule states that for a normal distribution: y y y
68% of the data will f all within 1 standard deviation of the mean 95% of the data will f all within 2 standard deviations of the mean Almost all (99.7%) of the data will f all within 3 standard deviations of the mean
Note that these values are appro ximations. For example, according to the normal curve probability density function, 95% of the data will fall within 1.96 standard deviations of the mean; 2 standard deviations is a convenient approximation. Normal Distribution and the C entral Limit Theor em
The normal distribution is a widely observed d istribution. Furthermore, it frequently can be applied to situations in which the dat a is distributed very differently. This extended applicability is possible because of the cent ra l limit theor em, which states that regardless of the distribution of the population, the distribution of the means of random samples approaches a normal distribution for a large sample size. A pplications to B usine ss Ad ministration
The normal distribution has applications in many a reas of business administration. For example: y
y y
Modern
portf olio theory commonly assumes that the returns of a diversif ied asset portf olio f ollow a normal distribution. In operations management, process variations of ten are normally distributed. In human resource management, employee perf ormance sometimes is considered to be normally distributed.
The normal distribution often is used to describe random variables, especially those having symmetrical, unimodal distributions. In many cases however, the normal distribution is only a rough approximation of the actual distribution. For example, the physical length o f a component cannot be negative, but the normal distribution extends indefinitely in both t he positive and negative directions. Nonetheless, the resulting errors may be neg ligible or within acceptable limits, allowing one to solve problems with sufficient accuracy by assuming a normal distribution.
