# Mean, simplified from 3D to 1D

• pangyatou

#### 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

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

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

Really appreciate.

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

pangyatou said:
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 $a_x$ ?