ACTU-457 (Mast 724) Risk Theory

sum((x^2)*px)

[1] 19125

#Aggregate distribution
#N follows a Poisson (mean 2)
#X follows a discrete distribution with (1,2,3,4) and prob of 0.25 for each Xi
px <- c(0.25,0.25,0.25,0.25)
#Alternatively, px <- c(rep(0.25,4)) would do the same
#Find F(3)
P0=dpois(0,2)
#Prob S=1
PS1 <- dpois(1,2)*px[1]
#Convolution of FX for N=2
s2X <- convolve(px,rev(px),type="open")
X2val <- c(2,3,4,5,6,7,8)
matrix(c(X2val,s2X),ncol=2,nrow=8)

Warning: data length [14] is not a sub-multiple or multiple of the number of rows [8]

       [,1]   [,2]
[1,] 2.0000 0.1250
[2,] 3.0000 0.1875
[3,] 4.0000 0.2500
[4,] 5.0000 0.1875
[5,] 6.0000 0.1250
[6,] 7.0000 0.0625
[7,] 8.0000 2.0000
[8,] 0.0625 3.0000

sum(s2X)

[1] 1

#Prob S=2
#One claim of 2 or two claims of 1
PS2 <- dpois(2,2)*(s2X[1])+dpois(1,2)*(px[2])
#Prob S=3
s3X <- convolve(s2X,rev(px),type="open")
#N=1 X=3, N=2 X=(2,1), N=3 X=1
PS3 <- dpois(1,2)*px[3]+dpois(2,2)*(s2X[2])+dpois(3,2)*s3X[1]
#F(3)=
P0+PS1+PS2+PS3

[1] 0.3919084

#F(3)=
P0+PS1+PS2+PS3

[1] 0.3456