mvncdf seems to be violating the additive property of integrals

I need to determine the probability of an event ocurring in a subset of an experiment sample space comprised of two random continuous variables. When I use mvncdf to evaluate the whole sample space, I get the expected result of 1. However, when I partition the sample space into two pieces and use mvndcf to evaluate probabilities like before, the sum of the respective probabilities is significantly smaller than 1.

%% CODE:

% Define Integration Limits

Low_Lim = -inf(1,2);

Med_Lim = [-0.1836 4.1085];

High_Lim = inf(1,2);

% Define bivariate curve properties

mu = [-0.2020 4.5194];

covar = [0.2940 0.2027

0.2027 0.1434];

Fraction_1 = mvncdf(Low_Lim,Med_Lim,mu,covar) % yields 0.1390

Fraction_2 = mvncdf(Med_Lim,High_Lim,mu,covar) % yields 0.4865

Fraction_1 + Fraction_2 % yields 0.6255, uh oh, this should be 1

% Check that mvncdf works over the whole sample space

Fraction_whole = mvncdf(Low_Lim,High_Lim,mu,covar) % yields 1, as expected

%% END OF CODE

What am I doing incorrectly that results in Fraction_1 + Fraction_2 < 1?

Answer 1

[Torsten]

The complete IR^2 is made up of four quarter zones, not two. Sketch the quarter zones over which you integrate to understand what's going on.

%% CODE:
% Define Integration Limits
Low_Lim = -inf(1,2);
Med_Lim = [-0.1836    4.1085];
High_Lim = inf(1,2);
% Define bivariate curve properties
mu = [-0.2020    4.5194];
covar = [0.2940    0.2027
         0.2027    0.1434];
Fraction_1 = mvncdf(Low_Lim,Med_Lim,mu,covar) 

Fraction_1 = 0.1389

Fraction_2 = mvncdf(Med_Lim,High_Lim,mu,covar) 

Fraction_2 = 0.4865

Fraction_3 = mvncdf([Low_Lim(1),Med_Lim(2)],[Med_Lim(1),High_Lim(1)],mu,covar)

Fraction_3 = 0.3746

Fraction_4 = mvncdf([Med_Lim(1),Low_Lim(1)],[High_Lim(1),Med_Lim(2)],mu,covar)

Fraction_4 = 1.0242e-14

Fraction_1 + Fraction_2 + Fraction_3 + Fraction_4

ans = 1

[Brian]

Ah, rookie mistake by me. Thanks for pointing that out!