Monarch point generation simulation - dutch republic & ottoman harem

Hi, as you all probably know, the Dutch Republic government type grants you a choice between two rulers every 4 years. In the new expansion, the Ottomans will be granted a choice between three different heirs. Assuming that each ruler's stats are generated independently of each other, it is common sense that if the player always chooses the ruler with the higher monarch points (let's say the ruler for whom the sum of his monarch points is greatest), then the player will have, on average, rulers with more monarch points.

I was thinking today while swimming in monarch points in my Netherlands game: on average, how much better are the rulers? I've heard on forums and read on the wiki that the number of points for each monarch power is generated as the sum of two discrete uniform random variables from 0 to 3. I tried for a while to come up with an analytical solution using probability but the computation was taking too long. I've been learning a little bit of programming recently so I decided to try my hand at running a simulation in matlab to find the mean monarch points per ruler in the normal case, for the Dutch Republic, and for the new Ottoman Harem:

clear all
tic
n=200000;
for i=1:n
    X(i,:)=randi(4,1,3)+randi(4,1,3)-2;
    Y(i,:)=randi(4,1,3)+randi(4,1,3)-2;
    %Ottoman harem - 3rd choice
    Z(i,:)=randi(4,1,3)+randi(4,1,3)-2;
    XSUM(i)=sum(X(i,:));
    YSUM(i)=sum(Y(i,:));
    %Ottoman harem - 3rd choice summed
    ZSUM(i)=sum(Z(i,:));  
end
%Dutch republic
for i = 1:n
    if YSUM(i) > XSUM(i);
        A(i) = YSUM(i);
    else 
        A(i) = XSUM(i);
    end
end
%Ottoman harem
for i = 1:n
    if YSUM(i) > XSUM(i) && YSUM(i) > ZSUM(i);
        B(i)=YSUM(i);
    elseif ZSUM(i) > XSUM(i) && ZSUM (i) > YSUM(i);
        B(i)=ZSUM(i);
    else
        B(i)=XSUM(i);
    end
end
%Sample means for sum of monarch points - regular case
sum(XSUM)/n
sum(YSUM)/n
sum(ZSUM)/n
%Sample means for dutch republic/ottoman harem
sum(A)/n
sum(B)/n
toc

The results were: Regular monarchy mean sum of monarch points: 8.9971

Dutch republic mean sum of monarch points: 10.5331

Ottoman Harem mean sum of monarch points: 11.1742

Some rough calculations:

Let's say that Mehmet dies in 1480 or so (this is a bit of a guess). If all subsequent rulers are generating 2.2 more monarch points per month in the new patch than in the current patch, then you will earn, on average, 2.2 * 12 * 340 = 9000 more monarch points over the course of the game - a pretty significant buff!

For the Dutch: supposing that you form Netherlands around 1520 (this is when I did in my recent game, I think you can do it earlier), then you have about 300 years of rulers generating 1.5 more monarch points per month, for a total of ~ 5400 monarch points extra compared to normal rulers.

Obviously this whole procedure relies on the assumption of how the monarch points are generated, the assumption that rulers to choose from are generated independently. It could also be an inaccurate measure of many more monarch points you will earn from 1444-1821, since as the Ottomans you may only have ~10-20 rulers over the course of the game, and you may not always choose the ruler for whom the sum of his monarch points is highest (he may have low admin power!). Also - let me know if you see anything wrong with this.

tl;dr ottoman rulers will have, in Rights of Man, on average 2.2 more monarch points than normal rulers, dutch rulers currently have about 1.5.