Finding the Expectations E(x) and E(y) for f(x,y,z) = 6e^{-x-y-z} for 0<x<y<z<infinity, 0 elsewhere

[mario99]

Let the joint probability density function of [imath]X, Y, \ \text{and} \ Z[/imath] be given by

[imath]\displaystyle f(x,y,z) = \begin{cases} 6e^{-x-y-z} & \ \ \ ,0< x < y < z < \infty \\ 0 & \ \ \ , \text{elsewhere} \end{cases} [/imath]


Find [imath]E(x) \ \text{and} \ E(y).[/imath]


Is it correct to write?

[imath]\displaystyle E(x) = \int\int\int 6xe^{-x-y-z} \ dx \ dy \ dz[/imath]

AND

[imath]\displaystyle E(y) = \int\int\int 6ye^{-x-y-z} \ dx \ dy \ dz[/imath]

Or do I have first to find the marginal joint probability density functions [imath]f_X(x) \ \text{and} \ f_Y(y)[/imath]?

 

[BigBeachBanana]

Let the joint probability density function of [imath]X, Y, \ \text{and} \ Z[/imath] be given by

[imath]\displaystyle f(x,y,z) = \begin{cases} 6e^{-x-y-z} & \ \ \ ,0< x < y < z < \infty \\ 0 & \ \ \ , \text{elsewhere} \end{cases} [/imath]


Find [imath]E(x) \ \text{and} \ E(y).[/imath]


Is it correct to write?

[imath]\displaystyle E(x) = \int\int\int 6xe^{-x-y-z} \ dx \ dy \ dz[/imath]

AND

[imath]\displaystyle E(y) = \int\int\int 6ye^{-x-y-z} \ dx \ dy \ dz[/imath]

Or do I have first to find the marginal joint probability density functions [imath]f_X(x) \ \text{and} \ f_Y(y)[/imath]?

Find marginal functions first.

 

[mario99]

Find marginal functions first.

I am sorry to tell you this again, but I am having a hard time to understand how to set the limits of integration when the domain is in series like:

[imath]0 < x < y < z < 1[/imath]

or

[imath]0 < x < y < z < \infty[/imath]


[imath]\displaystyle f_X(x) = \int\int f(x,y,z) \ dy \ dz[/imath]


[imath]\displaystyle f_Y(x) = \int\int f(x,y,z) \ dx \ dz[/imath]


Can you help me in setting the limits? Don't just set the limits, explain also how to set them. I want to understand the method so that I can apply it in all other problems by myself.