How to evaluate this probability

[GW008]

P={ z + x*y/(y+a) > x/(x*k+1)}
where x,y,z are exponentially distributed with mean xbar,ybar,zbar.
a,k are constants.
please suggest me how to evaluate this probability...
for k=0, P lies between 0 and 1 but for any positive value of k, P become higher than one which is unacceptable...
take xbar=1 ybar=3.14 zbar=1

 

[Stephen Tashi]

P={ z + x*y/(y+a) > x/(x*k+1)}
where x,y,z are exponentially distributed with mean xbar,ybar,zbar.
a,k are constants.

Your notation is unclear. Are you asking for the probability P of an event defined by the set:
{(x,y,z) such that z + x*y/(y+a) > x/(x*k + 1) } ?

for any positive value of k, P become higher than one which is unacceptable...
take xbar=1 ybar=3.14 zbar=1

How did you arrive at that conclusion?

 

[GW008]

Your notation is unclear. Are you asking for the probability P of an event defined by the set:
{(x,y,z) such that z + x*y/(y+a) > x/(x*k + 1) } ?

sorry for applying equal to sign, Yes u asking me right question

How did you arrive at that conclusion?

I try to solve this probability but I didn't get its close form so I plot it in MATLAB for different values of xbar.
But as I take the value of k greater than zero, Probability became larger than one...

Please suggest something

 

[Stephen Tashi]

so I plot it in MATLAB for different values of xbar.

"It"? Plot what? You aren't stating the problem clearly. What function did you plot?

 

[GW008]

"It"? Plot what? You aren't stating the problem clearly. What function did you plot?

"It refers to probability here.

probability of positive secrecy = P{ z + x*y/(y+a) > x/(x*k+1)}
where x,y,z are exponentially distributed with mean xbar,ybar,zbar.
a,k are constants.
please suggest me how to evaluate this probability...
for k=0, P lies between 0 and 1 but for any positive value of k, P become higher than one which is unacceptable...
ybar=3.14 zbar=1

I tried to solve this Probability to some extent but unable to express Probability in closed form. So, I plot this probability for different values of xbar(0-100). for k=0, this probability decrease from 1 to 0 but for k>0 this probability increase from 1 to 2.5 which is unacceptable...

I think now you might get what I am want to tell...

 

[Stephen Tashi]

"It refers to probability here.

If you are going to plot a function, "it" must be a function. What function are you plotting? What is its formula?

The general form of your question appears to be:

X,Y,Z are independent random variables
f(x,y,z), g(x,y,z) are given functions.
S is the event {(x,y,z) such that f(X,Y,Z) > g(X,Y,Z) }
Find P(S)

But what are you plotting? It wouldn't make sense to plot f(X,Y,Z) or g(X,Y,Z).

If you don't know the formula for P(S), how can you plot anything? You could run a Monte-Carlo simulation and estimate P(S), but that would never produce a number greater than 1.

 

[GW008]

Thanks for the above advise...

As event S has three random variables (all are independent) that means P(S) is a case of triple integration. I solve one integration but unable to evaluate other two integrations. In matlab, we can directly plot integral equation using function "integral , integral2, integral3". I use "integral2' to evaluate P(S) and plot it by varying xbar for different values of k...for k=0, this probability decrease from 1 to 0 but for k>0 this probability increase from 1 to 2.5.

It can be happened that i evaluate that first integration wrong or i did some other mistake... Please evaluate P(s)...
Thanks