在多元正态分布中找出概率

Question

I am using python. I know that to find probability in multivariate normal distribution I have to use following:

fx(x1,…,xk) = (1/√(2π)^k|Σ|) * exp(−1/2(x−μ)T* Σ^-1 *(x−μ))

where x = [x1, x2]

I have different values of x1 and x2.

but here I have to find probability for:

0.5< x1<1.5 and 4.5< x2<5.5

I know how to use this formula for single value of x1 and x2. But I am confused in this case. Please help.

Answer 1

What you need to so is find the area beneath the function for the rectangle bounded by 0.5 < x1 < 1.5 and 4.5 < x2 < 5.5 .

As a quick and dirty solution, you could use this code to do a two-variable Reimann sum to estimate the integral. A Reimann sum just divides the rectangle into small regions and approximates the area under each region as if the function was flat.

Provided you've defined your distribution as the function f .

x1Low   = 0.5
x1Hi    = 1.5
x2Low   = 4.5
x2Hi    = 5.5

x1steps = 1000
x2steps = 1000
x1resolution = (x1Hi-x1Low)/x1steps
x2resolution = (x2Hi-x2Low)/x2steps
area = x1resolution*x2resolution

x1vals = [x1Low + i*x1resolution for i in range(x1steps)]
x2vals = [x2Low + i*x2resolution for i in range(x2steps)]

sum = 0;
for i in range(len(x1vals-1)):
    for j in range(len(x2vals-1)):
        sum += area * f(x1vals[i],x2vals[j])

print sum

Keep in mind that this sum is only an approximation, and not a great one either. It will seriously over- or under-estimate the integral in regions where the function changes too quickly.

If you need more accuracy, you can try implementing triangle rule or simpsons's rule , or look into scipy's numerical integration tools .