matlab - Probabilities of bivariate normally distributed random variable -

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i'm trying calculate probability of bivariate normal distribution on specific area using matlab.

lets assume random variable follows standard normal distribution , want calculate mass of unit circle.

i used following code:

fun = @(x,y) exp(-0.5*(x.^2+y.^2))/(2*pi); ymin = @(x) -sqrt(1-(x.^2)); ymax = @(x) sqrt(1-(x.^2)); integral2(fun,-1,1,ymin,ymax)

i 0.3935. i'm wondering result correct.

can confirm result correct or point on mistake made?

i think correct. checks:

  • integrate on large square , see if result 1:

    >> integral2(fun,-5,5,-5,5) ans =    0.999998853581851

  • the 90-percentile of univariate gaussian distribution is

    >> norminv(.9) ans =    1.281551565544601

    so, integral of function on [−∞,∞] × [−∞,1.281] should 0.9:

    >> integral2(fun,-10,10,-10,norminv(.9)) ans =    0.900000750806316

  • the definitive, monte carlo check:

    >> n = 1e6; x = randn(1,n); y = randn(1,n); mean((x>-1)&(x<1)&(y>-sqrt(1-(x.^2)))&(y<sqrt(1-(x.^2)))) ans =    0.393678000000000

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