 Hi everyone, it's MJ and in this video we're looking at a sample question from exam P And this is a client spends X minutes in insurance agents waiting room So waiting room is X and Y minutes in the meeting with the agent So the meeting is Y the joint density function of X and Y can be modeled by and what we basically have here is a Joint exponential distribution So we almost I mean you don't have to separate it But you could think of separating it as follows where we have for X. We've got this one over here and Then for Y we have this over here now The interesting thing is is that what we're asked in this question is which are the following expressions? Represent the probability that a client spends less than 60 minutes at the agent's office And if we look at the multiple choice potential answers, we'll see that the only difference is The you know these limits over here for our integrals everything else is is the same Now the tricky part is or I think a lot of people might make a little bit of mistake is they're thinking oh look 40 and 40 for X and 20 and 20 for Y and they're thinking it's like this, but it's not necessarily the situation that might be You know the parameter that is used. That's the the waiting time rate But the question is saying what is the probability that the client spends less than 60 minutes at the agent's office? And is not giving us any indication of or saying, you know that's been so much and so much there Which means straight away we can actually cancel out all of the ones that are using the 40 and the 20 instead we can focus on These two that are looking at 60 minutes Now D is incorrect because it's not factoring in the sense that these These times are joint In the sense that the more time you spend in the one it's going to affect the time you spend in the other one so D can also be taken out which means by You know, this is the fact that he is the only one left it it must be the answer But it's always good to check and yes, you see that's what we are looking for in our joint distribution Is the fact that we want to get everything with regards to X? So we're gonna be putting the 60 minus X over here and then once we integrate that out with our X's We can then integrate out with just the X's and get and get our result so that is the one we are looking for and The way to work it out was that you know the total time is to be less than 60 minutes So if X minutes are being spent in the waiting room, you know in the range of 0 to 60 and from 0 to 60 Minus X minutes are spent in the meeting itself But yeah, it's a bit of a tricky one that for for joint distribution Functions, but if you've got any questions, let me know in the comment section below. Thanks for watching guys. Cheers