Difference between two vector CDFs

I am struggling to work out how to derive the vector for Prob(A - B > 0) where A and B are CDFs of independent variables in vector form.

I thought going through each point in the CDF vectors and multiplying 1-CDF_B by CDF_A would give the correct result, but the resulting vector doesn't sum to 1.

[Torsten]

What do you mean by "vector for Prob(A-B>0)" ?

In my opinion, Prob(A-B>0) is a single value, not a vector.

Best wishes

Torsten.

[Image Analyst]

Can you attach plots of A and B. Are they curves that go from (0,0) to 1,1) and cross each other some number of times, or what? Help us help you.

[Ulrik William Nash]

Here is a subplot showing (left) the PDFs and (right) CDF. These diagrams are have been generated by the values contained in four vectors.

My question is basically, how to I obtain (using MATLAB operations) another vector from the ones I have, which captures Prob(A > B)?

[Torsten]

Multiply PDF-vector for A with CDF-vector for B, sum the products and multiply the result by the distance of the points on the x-axis (deltax).

(Follows directly from the formula below).

Note that Prob(A-B>0) is not a vector, but a scalar value.

Best wishes

Torsten.

[Ulrik William Nash]

My apologies, Torsten. I haven't been very clear. What I am looking for is not a scalar, but the PDF or CDF for another random variable, let's call it C, which equals A - B. Then, when I have this vector, I can find A > B or equivalently, C > 0.

[Torsten]

The CDF of C=A-B at a point z can be obtained as follows:

Multiply PDF-vector of A at points x_i with CDF-vector of B at points x_i-z, sum the products and multiply the result by the distance of the points x_i (deltax). Then take the negative of this value and add 1.

The exact formula can be derived as follows :

F_C(z)
= Prob(A-B<=z) 
= integral_{x=-oo}^(x=+oo) Prob(A=x)*Prob(B>=x-z) dx
= integral_{x=-oo)^(x=+oo) f_A(x)*(1-F_B(x-z)) dx
= 1 - integral_{x=-oo}^{x=+oo} f_A(x)*F_B(x-z) dx

Maybe MATLAB's "conv" for the vectors f_A and F_B can automatically perform the task you are looking for, but I don't have the time to go into detail.

Best wishes

Torsten.

Answer 1

[Torsten]

Prob(A>B)
= integral_{a=-oo}^{a=+oo} Prob(A=a)*Prob(B<a) da 
= integral_{a=-oo}^{a=+oo} f_A(a)*F_B(a) da 
= integral_{a=-oo}^{a=+oo} (dF_A(a)/da)*F_B(a) da

where f_A, f_B denote PDFs of A and B and F_A, F_B denote CDFs of A and B.

Best wishes

Torsten.