Joint Normal CDF

#1
Hello

I am new to this forum as well as to actuary field. I have this question:

There are two random variables X_1 and X_2 following standard normal distributions:
X_1 ~ N(0,1)
X_2 ~ N(0,1)
If we now set
Y_1 = X_1
Y_2 = X_1+X_2
then we know that
Y_1 ~ N(0,1)
Y_2 ~ N(0,2)
The question is how to find the joint CDF Of Y = (Y_1, Y_2). Can we find the joint CDF in a closed form? If yes, how?

This is a problem arising in the field of risk claims.

Is there anyone who can guide me in resolving this issue?

Many thanks in advance

Kind regards
hpriye
 

vinux

Dark Knight
#2
If you know the joint CDF Of (X_1, X_2) then we can find the joint CDF of Y. Or are you assuming X_1 and X_2 are independent?

Vinux
 
#3
Many thanks!

Yes, indeed X_1 and X_2 are independent standard normal.

So what is the procedure for finding the joint CDF of Y=(Y_1, Y_2), where Y_1 = X_1 and Y_2 = X_1+X_2?

Kind regards
hpriye
 

vinux

Dark Knight
#4
Then it is not difficult.
In this case Y follows BVN(bivariate normal distribtion) with mean [ 0 0]' and Variance matrix
V(Y) = [ 1 1; 1 2]
( details: V(Y_1) = V(X_1) = 1 , V(Y_2) = V(X_1) + V(X_2) = 2
& Cov(Y_1,Y_2) = E{X_1*(X_1+X_2)} = E(X_1*X_1) + E(X_1)*E(X_2) = 1+0 =1 )

I think this solve the problem
------------------------------------
And if you wanted to calculate CDF at particular point EG: Y_1 = 1 and Y_2 = 4
i e X_1 = 1 & X2_1 = 4-1 = 3

so CDF Y(1,4) = CDF X(1,3) = CDF(1) * CDF(3)

Regards
Vinux aka Richie
 
#5
Hello

Whatever you have explained is for finding the joint probability distribution function (PDF) and not the cumulative distribution function (CDF)

The definition of CDF is (in the case of normal distribution)

doubleIntegral (PDF function, y_2=-infinity to v, y_1=-infinity to u);

... and what I am looking for is a closed form expression for the above integral which is perhaps impossible - as I have now realized.

.. But thanks for your efforts and time

Kind regards
hpriye
 

vinux

Dark Knight
#6
The double integreal part can be replaced by product of two single integral.Since X_1 and X_2 are independent
and Single integral ( Univariate cumilative normal probabilities ) are avaliable in most of the tools ( Excel Sas R etc)
Details:
If you wanted to convert CDF of Y then
First convert Y in terms of X then
CDF Y(1,4) = CDF X(1,3) = CDF(1) * CDF(3) ( see the example mentioned above)
Here CDF(1) = integral ( -inf to 1) std normal density) =~ 0.84
and CDF(3) =~ 0.99
so CDF Y(1,4) = 0.84*.99 = 0.83

Regards
Vinux aka Richie
 
#7
Hello

No, it is not possible to write it as product of two integrals for the PDF is :

1/(2*pi) exp(-1/2*y_1^2)*exp(-1/2*(y_2-y_1)^2)

to be integrated from -infinity to v and -infinity to u

Kind regards
hpriye
 

vinux

Dark Knight
#8
Hello

No, it is not possible to write it as product of two integrals for the PDF is :

1/(2*pi) exp(-1/2*y_1^2)*exp(-1/2*(y_2-y_1)^2)

to be integrated from -infinity to v and -infinity to u

Kind regards
hpriye
---------------------------
Yup.. you are true and thank you.. CDF of Y will be in the form of double integral. I think it can't be simplify further.