Show, by calculation, that the mean of y doesn't exist

#1
This is one of my questions for my homework, I have attached a copy of the function and also my attempt, I'm not sure if I've done it right. And if I have, why does the answer suggest that the mean of y doesn't exist? I understand that if the function is infinite, then it has no mean, however I don't understand why this answer would suggest that it has no mean?
 

BGM

TS Contributor
#2
First of all you need the marginal pdf [math] f_Y(y) [/math] by integrating out [math] x [/math] first, i.e.

[math] E[Y] = \int_0^{+\infty}\int_0^{+\infty} yf_{X,Y}(x,y)dxdy [/math]

Let you try first before discussing the remaining integration issue.
 

BGM

TS Contributor
#5
The marginal pdf you got in #3 is correct. If you are familiar with Gamma pdf you can very quickly to obtain that answer without integration by part.

It shows that the marginal pdf is a Pareto distribution, or the Lomax distribution with parameter [math] \alpha = \lambda = 1 [/math]

https://en.wikipedia.org/wiki/Lomax_distribution

You should continue with the marginal pdf and obtain the answer (Or if you know the survival function it could help too). Not quite follow what you did in the last post.
 
#6
Unfortunately we haven't learnt the other methods as of yet, so I've stuck with the marginal pdf approach. I've attached what I've done, and just wanted confirmation on whether I've done it right.

Thanks in advance!