Example R programs and commands
		23. Poisson density and tests for randomness

# All lines preceded by the "#" character are my comments.
# All other left-justified lines are my input.
# All other indented lines are the R program output.

# POISSON probability density
#
# Example: probability of x=5 counts when the mean is lambda=7.
x<-5           # event: count
lambda<-7      # situation: expected count
dpois(x, lambda)    # probability of count x when the mean is lambda

# Complete list of probabilities for x=0,1,2,...,(big):
dpois(0:20, lambda)    # probabilities of counts x=0,1,...,20

# Cumulative Poisson density function:
ppois(x, lambda)    # probability of x or fewer counts
ppois(x, lambda, lower.tail=FALSE)    # probability of more than x counts
1-ppois(x, lambda)                 # ...same result as the previous line



# POISSON randomness tests
#
# EG: test if the following frequency table has a Poisson density.
# Raisin count: 0     1     2     3     4     5     6 or more
# Frequency:    5    12    18    14     5     1     0
#
rcnt<-0:6       # Table contains frequencies of counts 0,1,...,6
freq<-scan()
 5    12    18    14     5     1    0

lasti<-length(rcnt)   # last index in the rcnt[] and freq[] arrays.

rcbar<- sum(rcnt*freq)/sum(freq);  rcbar  # average raisin count per slice
phat<-dpois(rcnt,lambda=rcbar) # Poisson density for the observed raisin counts
# ... piling the probabilities for all counts bigger than
#     the maximum count with a nonzero frequency into the last value:
phat[lasti] <- ppois(rcnt[lasti-1], lambda=rcbar, lower.tail=FALSE)

# NOTE: the expression finds the actual last count with nonzero frequency:
max(rcnt*(freq>0))

# Perform the goodness-of-fit test after correcting the number of
# degrees of freedom to be lasti-2:
pchisq(chisq.test(freq,p=phat)$statistic,df=lasti-2,lower.tail=FALSE) 
# OUTPUT:
#    X-squared
#    0.4656001
#    Warning message:
#    Chi-squared approximation may be incorrect in: chisq.test(freq, p = phat)
#
# The p-value is 0.4656001.
# Conclusion: DO NOT REJECT H0: raisins are randomly distributed by slice.


# Example with decidedly NON-RANDOM raisin distribution.
# Raisin count: 0     1     2     3     4     5     6 or more
# Frequency:    0     0     0     0     0     50    0
#
rcnt<-0:6       # Table contains frequencies of counts 0,1,...,6
freq<-scan()
    0     0     0     0     0     50    0

lasti<-length(rcnt)   # last index in the rcnt[] and freq[] arrays.

rcbar<- sum(rcnt*freq)/sum(freq);  rcbar  # average raisin count per slice
phat<-dpois(rcnt,lambda=rcbar) # Poisson density for the observed raisin counts
# ... piling the probabilities for all counts bigger than
#     the maximum count with a nonzero frequency into the last value:
phat[lasti] <- ppois(rcnt[lasti-1], lambda=rcbar, lower.tail=FALSE)

# Perform the goodness-of-fit test after correcting the number of
# degrees of freedom to be lasti-2:
pchisq(chisq.test(freq,p=phat)$statistic,df=lasti-2,lower.tail=FALSE)
# OUTPUT:
#      X-squared								
#      9.27622e-49							       
#      Warning message:							        
#      Chi-squared approximation may be incorrect in: chisq.test(freq, p = phat)
#
# The p-value is 9.27622e-49
# Conclusion: REJECT H0 for HA: raisins are NOT randomly distributed by slice.