834 
p*— — n 1 1 i 1 
.10 .20 .30 .40 .50 .60 .70 .80 .90 1.0 
POSITIVE PHOTOGRAPHIC DENSITY 2000 DEGREE DAY IMAGE 
FIGURE 5. The probability density functions for 5 crop 
types based upon the assumption of normality and the 
distribution parameters obtained from sample data. 
(Valid for purposes of this example only.) 
Compare Figure 1 with Figure 5 in order to visualize the conversion of 
raw descriptive data into crop-dependent probability density functions. 
The statistical framework of the probabilistic key is based upon 
Bayes Rule in the form: 
p(X ) P(T|X ) 
(1) P(X IT) = i — 
p(X.) P(T|x.) 
where; 
P(X^jT) is the conditional probability of crop type X given the set of 
photo-density values T; p(X_^) is the probability that a field is crop type 
X before considering any photo-densities, i.e., apriori probability of the 
crop type; 
and P(T|X_^) is equal to P(t^jX_^) P(t 2 |X^) where P(t^jX^)is the conditional 
probability of photo-density t^ on late spring imagery given X_^ and 
P(t 2 |X^ is the conditional probability of photo-density t^ on late summer 
imagery given the same X. 
The crop-dependent conditional probabilities P(t.|X„) are determined 
over the range of (t) for each crop type by normalizing the respective 
PDF and using a Z-score table (to fit the data to a normal distribution) 
to evaluate the resultant standardized photo-density values with respect 
to probability of occurence.