#########################################################################
## This function returns the value of the probability density function ##
## of a mixed Poisson distributed random variable with the parameters  ##
## given in exercise 1, sheet #5                                       ##
#########################################################################

pdf_N <- function(k) {
	lambda <- c(60,35,16,12,6)
	p <- c(0.05,0.2,0.4,0.25,0.1)
	sum <- NULL
	for (i in 1:length(k)) {
		temp <- exp(-lambda)*lambda^k[i]/factorial(k[i])
		sum <- c(sum,t(p)%*%temp)
	}
	return(sum)
}

## Example 1: value of the pdf at k=1
pdf_N(1)

## Example 2: value of the pdf at k=1,3,5
k <- c(1,3,5)
pdf_N(k)


## (a)

lambda <- c(60,35,16,12,6)
p <- c(0.05,0.2,0.4,0.25,0.1)
mixed_mean <- t(lambda) %*% p
mixed_mean

## (b)

## This function returns the value of the cdf of the mixed Poisson distributed rv
cdf_N <- function(k) {
	if (!is.vector(k)) {
		cat("The argument for cdf_N has to be a vector.")
	} else {
		values <- NULL
		for (i in 1:length(k)) {
			x <- 0:k[i]
			temp <- sum(pdf_N(x))
			values <- c(values, temp)
		}
		return(values)
	}
}

k <- c(10,14,20,30,40,60)
cdf_N(k)
ppois(k, 20)