## sheet 6, ex. 3 (a)

p <- 1/117411*c(103705,11632,1767,255,44,6,2)
k <- 0:6

ev <- p %*% k
variance <- (k-ev)^2 %*% p

ev 
variance

## Since the expected value is less than the variance of the
## empirical distribution function, the sample is probably
## not Poisson distributed. 


## ex. 3 (b)

p
dpois(k, ev)
dpois(k, variance)


## ex. 3 (c)

pnegbinom <- function(k, alpha, p) {
	x <- (1-p)^alpha
	if (k == 0) {
		return(x)
	}
	a <- p
	b <- (alpha-1)*a
	for (i in 1:k) {
		x <- (a+b/i)*x
	}
	return(x)
}

for (i in 0:6) {
	cat(pnegbinom(i, 0.625507, 0.180254)," ")
}