How to Simplify the Mean of 3D Variables to 1D?

[pangyatou]

Hi,

There are three variables ax, ay and az, my question is:
How to simplify the mean value <(ax^2+ay^2+az^2)^(1/2)> to <|ax|> ?
What assumptions are required during the simplification?

The statistical property of ax, ay and az is <ax^2>=<ay^2>=<az^2>.
The assumption of the propability is: pdf(ax), pdf(ay) and pdf(az) are independent to each other: p(ax,ay,az)=p(ax)p(ay)p(az)

Thanks

 

[mathman]

No further assumptions are needed to carry out the calculation. It is messy because you are taking the square root before calculating the integral.

 

[pangyatou]

Thanks Mathman!
What theory or property can be applied to this problem? I don't even have a clue.

Really appreciate.

 

[mathman]

It is a 3-d integral where the integrand is the product of the 3 density functions multiplied by the expression (square root etc.).

 

[Stephen Tashi]

Hi,

There are three variables ax, ay and az, my question is:
How to simplify the mean value <(ax^2+ay^2+az^2)^(1/2)> to <|ax|> ?
What assumptions are required during the simplification?

Are you asking if the mean value of $r = \sqrt{a_x^2 + a_y^2 + a_z^2}$ must be equal to the mean value of the absolute value of $a_x$ ?