Probability distributions may be discrete or continuous. This week we will examine two common discrete distributions: the binomial and Poisson. We will use JMP to generate random samples from these distributions and explore their characteristics.

The binomial distribution is one of the most commonly encountered discrete probability distributions in biology. It is based on nominal scale data that come from a population with only two categories. One of the two categories is arbitrarily referred to as a "success" and the other a "failure" (based on which you are looking for) and these categories are mutually exclusive (i.e. male and female, black and white, left and right). The process of selecting an individual at random from the population is called a trial. The probability of a success remains constant from trial to trial. Furthermore, the outcome of any particular trial is not affected by the outcome of any other trial (i.e. trials are independent). The terms of the equation for the binomial distribution are calculated automatically by the computer using the equation:

In this equation P(X) is the probability of seeing X successes, n is the number of trials, p is the proportion of total occurrences that are successes, and q is the proportion that are failures (since p and q are mutually exclusive, q = 1 - p). Another term commonly encountered in binomial calculations is N, the number of observations.

The binomial distribution has several convenient mathematical features. First, the variance (s2) = n x p x q (the standard deviation is the square root of this). Second, the mean  (m) = n x p. Third, for large numbers of trials (perhaps n > 25), the shape of the binomial distribution is very similar to a normal distribution, allowing us to estimate probabilities without repeatedly using the cumbersome formula given above.

Another discrete probability distribution commonly encountered in biology is the Poisson distribution. This distribution is important in describing random occurrences of objects in space or events in time. The occurrence of an object or event is assumed to have no effect on the probability of a second occurrence of the same object or event (i.e. objects or events are independent). The formula for calculating the probability of an occurrence is:

In this equation, P(X) is the probability of seeing X successes, m is the mean number of occurrences, and e is a constant (2.718..., the base for natural logarithms). An interesting property of the Poisson distribution is that the mean and variance of the number of events per interval are equal. Thus for a Poisson distribution, the variance to mean ratio, often called the coefficient of dispersion (s2 /) = 1, which is an indication that events are randomly spaced. If this is not the case, the distribution is not a Poisson. A uniform distribution, for instance, has a variance/mean ratio less than 1, while a clumped distribution produces a ratio greater than 1.

When there are a large number of trials, and p is very small, the Poisson distribution is very similar in shape to the binomial distribution. When the mean is large the normal distribution approximates the Poisson distribution.

To experiment with these distributions you will need to use the calculator functions of JMP. Start by opening a new file (double click on the blank page icon), then setup four columns. To format a column to illustrate a probability distribution, select the column, then choose Formula from the pull-down menu for columns. The program will open up a new window, the calculator. This platform allows you to create complex formulas to produce the data for a variable. For today’s exercise we will concentrate on the random number generating functions of this calculator and on the probability functions.

In the calculator window, click on random from the set of choices to the right of the keypad. This will produce a set of terms in the far right window that will allow us to generate samples from a wide variety of probability distributions.

1. Format column 1 as a binomial distribution by choosing binomial after you have selected the random number generator (it’s probably worth changing the column name to binomial just for clarity’s sake). As described above, two parameters, the number of trials and the probability of success characterize the binomial. The calculator requires these values to be selected before it can generate a binomial distribution. Let’s assume that we wish to examine the distribution of male and female offspring in rats with litter sizes of 10. Click to highlight the first empty square in the formula window. This is where you select the number of trials. For this example choose 10 (the number of offspring in each litter). Select the second empty box, the one for probability. Again, for this example, where males and females are equally likely, set the probability of success to 0.5, then click OK. This will close the calculator window. If you now add 10 rows to your column you should obtain a random sample of 10 values from this particular binomial distribution.

2. Set up a new column and label it Poisson. As you did in the previous question, call up the calculator window and format this column as a Poisson distribution by choosing Poisson after you have selected the random function. In this case, we could be examining the number of Asian Gypsy moths found in insect traps throughout the lower mainland. For this distribution you must set one characteristic, the mean (equivalent to lambda in this instance). Set this value to 0.5 for now, meaning that on average we found one insect in every second trap.