4 assignments in Business Statistics due dates listed within description.

Discrete Probability Distributions

Probability Distribution

What is a probability distribution?  It is a listing of all the outcomes of an experiment and the probability associated with each outcome.

Characteristics of a Probability Distribution:

1. The probability of a particular outcome is between 0 and 1 inclusive.

2. The outcomes are mutually exclusive events.

3. The list is exhaustive, so the sum of the probabilities of the various events is equal to 1.

Try this experiment:

Randomly select a number from 1 to 4.  Now, do it again.  What are the possible ways of picking a 1?  They are: zero ones, one ones, and two ones.  What is the distribution of the number of ones?

Figure 1

Figure 2

The mean of the probability distribution is also its expected value.  To find the mean, you multiply the number of 1s time the probability and then add those numbers.

To find the mean:

· (0)(0.5625) = 0

· (1)(0.375) = 0.375

· (2)(0.0625) = 0.125

Add 0 + 0.375 + 0.125 = 0.5 to find the mean, or expected value.

To find the variance, you follow these steps.

1. Subtract the mean from each value, and square the difference.

2. Multiply each squared difference by its probability.

3. Sum the resulting products to arrive at the variance.

Figure 3

When you add the numbers in the last column, your total is 0.375, which is the variance.

To find the standard deviation, take the positive square root of the variance.  The standard deviation is 0.61237.

Random Variables

A random variable is a quantity resulting from an experiment that, by chance, can assume different values.

A discrete random variable can assume only certain clearly separated values.  It is usually the result of counting something.  Examples:  the number of people in a park or the number of loans approved by a bank.

A continuous random variable can assume an infinite number of values within a given range.  It is usually the result of some type of measurement.  Examples:  the weight of each watermelon picked on a certain day or the temperature outside as you are reading these notes.

The Binomial Probability Distribution

Characteristics of a Binomial Probability Distribution:

1. There are only two possible outcomes on a particular trial of an experiment.

2. The outcomes are mutually exclusive.

3. The random variable is the result of counts.

4. Each trial is independent of any other trial.

Binomial probability represents situations in which there can be only two outcomes.  For example, tossing a coin yields only two outcomes: a head or a tail, ignoring the minuscule probability that the coin could actually land on its edge.  The probabilities associated with binomial probability events can be calculated using a formula or using the binomial probability tables.  The following example illustrates the formula and the concepts underlying the tables for single and cumulative binomial probability events.

We wish to find the probability that exactly 2 heads will result from 5 tosses of a fair coin.

· n = 5

· r = 2

· P = 0.5

· q = 1 - 0.5 = 0.5

The formula yields the result:

Figure 4

From the multiplication law of probability, each 2-head event has a probability of:

(0.5)2 (0.5)3 = 0.55 = 0.03125 Figure 5

Using this, we can determine the complete probability distribution for the number of heads obtained from 5 tosses of a coin:

Figure 6

The cumulative probability of certain events also can be determined from the data. For example, what is the probability of getting less than 3 heads on 5 tosses of a fair die?

This also can be stated as "no more than 2 heads," which is the probability of 0 heads + the probability of 1 head + the probability of 2 heads = 0.3125 + .15625 + .31250 = 0.50

If you go to appendix A, for n = 5, r = 2, the cumulative probability table gives you a result of 0.50, corroborating this result.

Please visit the  Binomial Probability "Exactly" Website  and work through the additional binomial probability problems.

Hypergeometric probability distribution

We now look at how to address sampling when it is done without replacement.  The critical difference here between a hypergeometric and a binomial experiment is whether the probability of success on each trial is constant.  For binomial experiments, it is required to be so.  For hypergeometric, it is not, particularly in cases in which a population is not that large, without replacement, and the probability for each observation is going to change.  The hypergeometric distribution will be applied when a sample has been selected from a population that is finite and without replacement.  The second criterion is that the size of the sample is more than 5% of the total population.

A key formula for hypergeometric probability distributions:

P(x) = (SCx)(N-SCn-x) / NCn

Two properties that will be helpful to have as reference with respect to hypergeometric distributions are:

the mean = n * k / N

and the variance = n * k * (N – k) * (N – n) / [N2 * (N – 1)].

Please visit the  Hypergeometric Distribution Website  for a helpful tutorial on hypergeometric probability distributions.

Poisson Probability Distributions

One of the critical resources that businesses have to manage is time.  This is where Poisson probability distribution comes in.  There are several key characteristics of a Poisson experiment.  Outcomes must be classifiable as success or failure.  The next characteristics use the term region.  As they apply to Poisson distributions, regions may be defined as a period of time, an area, a volume, or a length (StatTrek.com, 2012).  For our use in business, the Poisson probability distributions will most likely apply to the number of events between two moments in time, such as the start of a clearance sale and the end of the sale.  The average quantity of successes is a known variable.  The next assumption is that the length of the interval and the probability are proportional.  The next assumption requires independent intervals.  The last is that the likelihood that an extremely small region containing the success is virtually zero (StatTrek.com, 2012).  Statisticians also casually refer to the Poisson probability distribution as the “law of improbable events.”

The textbook outlines three critical characteristics that each Poisson experiment will contain.  1) The random variable is the quantity of an event occurrences during the specified interval, 2) proportionality between the event the size interval, and 3) not only are the intervals independent, but they also do not overlap.  As you can see, there is some disparity in how Poisson distributions are viewed.  These differences will not affect your assignments for this module.  Please refer to page 208 for the formulas you will need for this module.

Please visit the  Poisson Distribution Website  and work through the additional example of the Poisson probability distribution in the tutorial.