Objective Problem Description Poisson Distribution; Poisson Moments Poisson Probabilities Poisson Quantiles
Compute probabilities and quantiles for a Poisson random variable representing the number of horse kicks and visualize these quantities using the Poisson Applet.
The number of horse-kick deaths of Prussian military personnel were recorded for each of 14 corps in each of 20 years from 1875-1894. The mean number of deaths per corps per year was 0.7. Four of the corps were atypical and were removed. The mean number of deaths per corps per year for the remaining corps was 0.6. The objective of this example is to compare the distributions before and after removing the suspect corps. We will assume the Poisson distribution holds for both scenarios with means given by the above sample means.
Examine the Poisson distribution of the horse-kick deaths when the mean (m) is 0.7. Notice the distribution is skewed to the right (positively skewed). This will always be the case for small m.
Two mutually exclusive button items are available: f(x) for computing probabilities and F(x) for computing quantiles. Initially, the graph is set to compute probabilities. A popup menu Prob is positioned in the top panel. It allows the type of probability calculation to be changed, e.g., P(a <= x <= b) can be selected.
The mean is a more meaningful measure of location than the median for the Poisson since it is the natural parameter of the Poisson. Also, the dispersion, as measured by the variance, is equal to the mean for a Poisson distribution, i.e., the value of the parameter m is both the mean and the variance.
We will now explore the effect on the Poisson distribution of deleting the 4 above-mentioned corps. Verify that m is 0.7. Choose X = ? from the Prob popup menu and set X to 0. In this case, P( X = 0) = 0.4965.
As m is decreased to 0.6 (you will need to double click on m, set increment to 0.1, and use the animation tool), note that P( X = 0) increases to 0.5488. The occurs because as the mean gets smaller, the probability of small values, e.g., 0, increases. Explore what happens to the shape of the distribution as m decreases (You may need to Rescale the distribution).
The above discussion implies that the probability of getting larger values must decrease as the mean decreases. Consider P( X >= 2). When m = 0.7, P(X >= 2) = 0.1558. This probability decreases to 0.1219 when m = 0.6.
In what way are the four deleted corps different from the remaining 10? It appears that the four corps had higher probabilities of more deaths in comparison with the remaining 10 corps. This is also reflected by a higher mean for the 14 corps as compared to that of 10 corps.
Suppose we want to know the quartiles for the two Poisson distributions. These are obtained from the cumulative distribution function F(x), i.e, by selecting the F(x) button item. First change m to 0.7 and then click on F(x). By default, q = 0.5 which corresponds to the median (the 0.5 quantile). The median is 1 and is given by the quantile equation in the tool bar and in the plot by inversely solving for x in F(x) = 0.5, i.e, follow the red lines from F(x) = 0.5 to x = 1.
Change q to 0.25 by pressing on the left triangle in the q animation tool. We see that the lower quartile is 0. Likewise, the upper quartile is found by pressing on the right triangular button until q is 0.75. The upper quartile is 1, the same as the median.
When m is reduced to 0.6 the median changes to 0, whereas the upper quartile is 1 and the lower quartile is 0. Thus the lower and upper quartiles do not change, but the median drops from 1 to 0. The quartiles do not change as a result of the discreteness of the distribution in comparison with the rather small change in m. The median changes by catching a jump in the probability that X = 0 when m changes from 0.7 to 0.6.