# Question:Evaluating double integrals: Calculating the probability (inequality relation) of joint, continuous (exponential) independent variables

## Question:Evaluating double integrals: Calculating the probability (inequality relation) of joint, continuous (exponential) independent variables

Maple

Hi

I am trying to follow a textbook example concerning the calculations of probability based on continous joint distributions. I cannot calculate the same result using Maple, specifically the double integral.

In a effort to determine what I need help with I have presented the example and what I know, in order to potentially rule out potential misconceptions with regard to the theory itself and maybe not a lack of Maple skills..

The example is a follows;

My thoughts / attempt:

So we are dealing with independent variables which are exponetially distributed. We need to find the prob. that P(X<Y), so we need to find the joint density of the distributions. We know the density of an exponential distributed variable, and since they are independent, the product of their densities is the desired joint density function f(x,y) we need in order to evaluate the probabilty;

My book states that the probability of a set B, w.r.t to two continuous distributions is

so with regard to my specific case, B can be substituted with X<Y, and as such should also be applied appropriately to the limits of the integrals.

We know that both distributions have the same exponential distribution (in) [0,+inf], and that X<Y is to be determined, thus we can conlude that

{(x,y) : 0 < x < y < +inf}

Thus the probability is given by (as presented in the book):

Can Maple directly solve this integral from such an expression?

The book chooses to split up the double integral with respect to the limits of the variables(distributions?);

x is of course the lowerbound for the dy integral as y is specfied to be larger than x..

So ultimately my problem is that I cannot replicate the last integral expression ,I end up with;

So an additional term.. I am just ignorant, and should I ultimately know that I have to disregard the contribution term containg the y variable seeing that we have to inetgrate w.r.t to the last dx integral?doubleintegralprobability.mw

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